{"id":202,"date":"2017-10-27T16:29:08","date_gmt":"2017-10-27T16:29:08","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/chapter\/projectile-motion\/"},"modified":"2017-11-08T03:24:00","modified_gmt":"2017-11-08T03:24:00","slug":"projectile-motion","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/chapter\/projectile-motion\/","title":{"raw":"Projectile Motion","rendered":"Projectile Motion"},"content":{"raw":"\n<div class=\"textbox learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<ul>\n<li>Identify and explain the properties of a projectile, such as acceleration due to gravity, range, maximum height, and trajectory.<\/li>\n<li>Determine the location and velocity of a projectile at different points in its trajectory.<\/li>\n<li>Apply the principle of independence of motion to solve projectile motion problems.<\/li>\n<\/ul>\n<\/div>\n<p id=\"import-auto-id1846742\"><span data-type=\"term\" id=\"import-auto-id1560216\">Projectile motion<\/span> is the <span data-type=\"term\" id=\"import-auto-id1846113\">motion<\/span> of an object thrown or projected into the air, subject to only the acceleration of gravity. The object is called a <span data-type=\"term\" id=\"import-auto-id1809247\">projectile<\/span>, and its path is called its <span data-type=\"term\" id=\"import-auto-id1397020\">trajectory<\/span>. The motion of falling objects, as covered in <a href=\"\/contents\/68db177b-ab5c-48b9-8bcc-721e140fed8b@2\">Problem-Solving Basics for One-Dimensional Kinematics<\/a>, is a simple one-dimensional type of projectile motion in which there is no horizontal movement. In this section, we consider two-dimensional projectile motion, such as that of a football or other object for which <span data-type=\"term\" id=\"import-auto-id1230666\">air resistance<\/span> <em data-effect=\"italics\"><em data-effect=\"italics\">is negligible<\/em><\/em>.<\/p>\n<p id=\"import-auto-id1696126\">The most important fact to remember here is that <em data-effect=\"italics\"><em data-effect=\"italics\">motions along perpendicular axes are independent<\/em><\/em> and thus can be analyzed separately. This fact was discussed in <a href=\"\/contents\/21d0e217-d50f-4901-af75-905e738eb4c4@4\">Kinematics in Two Dimensions: An Introduction<\/a>, where vertical and horizontal motions were seen to be independent. The key to analyzing two-dimensional projectile motion is to break it into two motions, one along the horizontal axis and the other along the vertical. (This choice of axes is the most sensible, because acceleration due to gravity is vertical\u2014thus, there will be no acceleration along the horizontal axis when air resistance is negligible.) As is customary, we call the horizontal axis the <em data-effect=\"italics\"><em data-effect=\"italics\">x<\/em><\/em>-axis and the vertical axis the <em data-effect=\"italics\"><em data-effect=\"italics\">y<\/em><\/em>-axis. <a href=\"#import-auto-id2242290\" class=\"autogenerated-content\">(Figure)<\/a> illustrates the notation for displacement, where [latex]\\mathbf{s}[\/latex] is defined to be the total displacement and [latex]\\mathbf{x}[\/latex] and [latex]\\mathbf{y}[\/latex] are its components along the horizontal and vertical axes, respectively. The magnitudes of these vectors are <em data-effect=\"italics\"><em data-effect=\"italics\">s<\/em><\/em>, <em data-effect=\"italics\"><em data-effect=\"italics\">x<\/em><\/em>, and <em data-effect=\"italics\"><em data-effect=\"italics\">y<\/em><\/em>. (Note that in the last section we used the notation [latex]\\mathbf{A}[\/latex] to represent a vector with components [latex]{\\mathbf{A}}_{x}[\/latex] and [latex]{\\mathbf{A}}_{y}[\/latex]. If we continued this format, we would call displacement [latex]\\mathbf{s}[\/latex] with components [latex]{\\mathbf{s}}_{x}[\/latex] and [latex]{\\mathbf{s}}_{y}[\/latex]. However, to simplify the notation, we will simply represent the component vectors as [latex]\\mathbf{x}[\/latex] and [latex]\\mathbf{y}[\/latex].)<\/p>\n<p id=\"import-auto-id1576953\">Of course, to describe motion we must deal with velocity and acceleration, as well as with displacement. We must find their components along the <em data-effect=\"italics\"><em data-effect=\"italics\">x<\/em><\/em>- and <em data-effect=\"italics\"><em data-effect=\"italics\">y<\/em><\/em>-axes, too. We will assume all forces except gravity (such as air resistance and friction, for example) are negligible. The components of acceleration are then very simple: <\/p>\n<p>[latex]{a}_{y}=\u2013g=\u20139.80 m{\\text{\/s}}^{2}[\/latex]. (Note that this definition assumes that the upwards direction is defined as the positive direction. If you arrange the coordinate system instead such that the downwards direction is positive, then acceleration due to gravity takes a positive value.) Because gravity is vertical, <\/p>\n<p>[latex]{a}_{x}=0[\/latex]. Both accelerations are constant, so the kinematic equations can be used.<\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id1767845\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Review of Kinematic Equations (constant [latex]a[\/latex])<\/div>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-891\">[latex]x={x}_{0}+\\stackrel{-}{v}t[\/latex]<\/div>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-557\">[latex]\\stackrel{-}{v}=\\frac{{v}_{0}+v}{2}[\/latex]<\/div>\n<div data-type=\"equation\" class=\"equation\">[latex]v={v}_{0}+\\text{at}[\/latex]<\/div>\n<div data-type=\"equation\" class=\"equation\">[latex]x={x}_{0}+{v}_{0}t+\\frac{1}{2}{\\text{at}}^{2}[\/latex]<\/div>\n<div data-type=\"equation\" class=\"equation\">[latex]{v}^{2}={v}_{0}^{2}+2a\\left(x-{x}_{0}\\right)\\text{.}[\/latex]<\/div>\n<\/div>\n<div class=\"bc-figure figure\" id=\"import-auto-id2242290\">\n<div class=\"bc-figcaption figcaption\">The total displacement [latex]\\mathbf{s}[\/latex] of a soccer ball at a point along its path. The vector [latex]\\mathbf{s}[\/latex] has components [latex]\\mathbf{x}[\/latex] and [latex]\\mathbf{y}[\/latex] along the horizontal and vertical axes. Its magnitude is [latex]s[\/latex], and it makes an angle [latex]\\theta [\/latex] with the horizontal.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1402609\" data-alt=\"A soccer player is kicking a soccer ball. The ball travels in a projectile motion and reaches a point whose vertical distance is y and horizontal distance is x. The displacement between the kicking point and the final point is s. The angle made by this displacement vector with x axis is theta.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_04_01.jpg\" data-media-type=\"image\/jpg\" alt=\"A soccer player is kicking a soccer ball. The ball travels in a projectile motion and reaches a point whose vertical distance is y and horizontal distance is x. The displacement between the kicking point and the final point is s. The angle made by this displacement vector with x axis is theta.\" width=\"350\"><\/span><\/p><\/div>\n<p id=\"eip-36\">Given these assumptions, the following steps are then used to analyze projectile motion:<\/p>\n<p id=\"eip-822\"><em data-effect=\"italics\"><strong>Step 1.<\/strong><\/em><em data-effect=\"italics\">Resolve or break the motion into horizontal and vertical components along the x- and y-axes.<\/em> These axes are perpendicular, so <\/p>\n<p>[latex]{A}_{x}=A\\phantom{\\rule{0.25em}{0ex}}\\text{cos}\\phantom{\\rule{0.25em}{0ex}}\\theta [\/latex] and <\/p>\n<p>[latex]{A}_{y}=A\\phantom{\\rule{0.25em}{0ex}}\\text{sin}\\phantom{\\rule{0.25em}{0ex}}\\theta [\/latex] are used. The magnitude of the components of displacement <\/p>\n<p>[latex]\\mathbf{s}[\/latex] along these axes are [latex]x[\/latex] and <\/p>\n<p>[latex]\\mathrm{y.}[\/latex] The magnitudes of the components of the velocity [latex]\\mathbf{v}[\/latex] are <\/p>\n<p>[latex]{v}_{x}=v\\phantom{\\rule{0.25em}{0ex}}\\text{cos}\\phantom{\\rule{0.25em}{0ex}}\\theta [\/latex] and <\/p>\n<p>[latex]{v}_{y}=v\\phantom{\\rule{0.25em}{0ex}}\\text{sin}\\phantom{\\rule{0.25em}{0ex}}\\mathrm{\\theta ,}[\/latex] where <\/p>\n<p>[latex]v[\/latex] is the magnitude of the velocity and [latex]\\theta [\/latex] is its direction, as shown in <a href=\"#import-auto-id1815222\" class=\"autogenerated-content\">(Figure)<\/a>. Initial values are denoted with a subscript 0, as usual.<\/p>\n<p id=\"eip-205\"><em data-effect=\"italics\"><strong>Step 2.<\/strong><\/em><em data-effect=\"italics\">Treat the motion as two independent one-dimensional motions, one horizontal and the other vertical.<\/em> The kinematic equations for horizontal and vertical motion take the following forms:\n<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-338\">[latex]\\text{Horizontal Motion}\\left({a}_{x}=0\\right)[\/latex]<\/div>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-362\">[latex]x={x}_{0}+{v}_{x}t[\/latex]<\/div>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-627\">[latex]{v}_{x}={v}_{0x}={v}_{x}=\\text{velocity is a constant.}[\/latex]<\/div>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-293\">[latex]\\text{Vertical Motion}\\left(\\text{assuming positive is up}\\phantom{\\rule{0.25em}{0ex}}{a}_{y}=-g=-9.\\text{80}{\\text{m\/s}}^{2}\\right)[\/latex]<\/div>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-131\">[latex]y={y}_{0}+\\frac{1}{2}\\left({v}_{0y}+{v}_{y}\\right)t[\/latex]<\/div>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-305\">[latex]{v}_{y}={v}_{0y}-\\text{gt}[\/latex]<\/div>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-542\">[latex]y={y}_{0}+{v}_{0y}t-\\frac{1}{2}{\\mathrm{gt}}^{2}[\/latex]<\/div>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-243\">[latex]{v}_{y}^{2}={v}_{0y}^{2}-2g\\left(y-{y}_{0}\\right)\\text{.}[\/latex]<\/div>\n<p id=\"eip-708\"><em data-effect=\"italics\"><strong>Step 3.<\/strong><\/em><em data-effect=\"italics\"> Solve for the unknowns in the two separate motions\u2014one horizontal and one vertical.<\/em> Note that the only common variable between the motions is time [latex]t[\/latex]. The problem solving procedures here are the same as for one-dimensional <span data-type=\"term\">kinematics<\/span> and are illustrated in the solved examples below.<\/p>\n<p id=\"eip-979\"><em data-effect=\"italics\"><strong>Step 4.<\/strong><\/em><em data-effect=\"italics\">Recombine the two motions to find the total displacement<\/em>[latex]\\mathbf{\\text{s}}[\/latex]<em data-effect=\"italics\"> and velocity <\/em>[latex]\\mathbf{\\text{v}}[\/latex]. Because the <em data-effect=\"italics\">x<\/em> - and <em data-effect=\"italics\">y<\/em> -motions are perpendicular, we determine these vectors by using the techniques outlined in the <a href=\"\/contents\/b9739bfd-dc9d-4f0a-b037-dc22884d30f3@10\">Vector Addition and Subtraction: Analytical Methods<\/a> and employing [latex]A=\\sqrt{{A}_{x}^{2}+{A}_{y}^{2}}[\/latex]<br>\n        and [latex]\\theta ={\\text{tan}}^{-1}\\left({A}_{y}\/{A}_{x}\\right)[\/latex] in the following form, where [latex]\\theta [\/latex] is the direction of the displacement [latex]\\mathbf{s}[\/latex] and [latex]{\\theta }_{v}[\/latex] is the direction of the velocity [latex]\\mathbf{v}[\/latex]:\n<\/p>\n<p id=\"eip-245\"><strong>Total displacement and velocity<\/strong><\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-743\">[latex]s=\\sqrt{{x}^{2}+{y}^{2}}[\/latex]<\/div>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-373\">[latex]\\theta ={\\text{tan}}^{-1}\\left(y\/x\\right)[\/latex]<\/div>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-679\">[latex]v=\\sqrt{{v}_{x}^{2}+{v}_{y}^{2}}[\/latex]<\/div>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-264\">[latex]{\\theta }_{v}={\\text{tan}}^{-1}\\left({v}_{y}\/{v}_{x}\\right)\\text{.}[\/latex]<\/div>\n<div class=\"bc-figure figure\" id=\"import-auto-id1815222\">\n<div class=\"bc-figcaption figcaption\">(a) We analyze two-dimensional projectile motion by breaking it into two independent one-dimensional motions along the vertical and horizontal axes. (b) The horizontal motion is simple, because [latex]{a}_{x}=0[\/latex] and [latex]{v}_{x}[\/latex] is thus constant. (c) The velocity in the vertical direction begins to decrease as the object rises; at its highest point, the vertical velocity is zero. As the object falls towards the Earth again, the vertical velocity increases again in magnitude but points in the opposite direction to the initial vertical velocity. (d) The <em data-effect=\"italics\">x<\/em> - and <em data-effect=\"italics\">y<\/em> -motions are recombined to give the total velocity at any given point on the trajectory.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id2275311\" data-alt=\"In part a the figure shows projectile motion of a ball with initial velocity of v zero at an angle of theta zero with the horizontal x axis. The horizontal component v x and the vertical component v y at various positions of ball in the projectile path is shown. In part b only the horizontal velocity component v sub x is shown whose magnitude is constant at various positions in the path. In part c only vertical velocity component v sub y is shown. The vertical velocity component v sub y is upwards till it reaches the maximum point and then its direction changes to downwards. In part d resultant v of horizontal velocity component v sub x and downward vertical velocity component v sub y is found which makes an angle theta with the horizontal x axis. The direction of resultant velocity v is towards south east.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_04_02.jpg\" data-media-type=\"image\/jpg\" alt=\"In part a the figure shows projectile motion of a ball with initial velocity of v zero at an angle of theta zero with the horizontal x axis. The horizontal component v x and the vertical component v y at various positions of ball in the projectile path is shown. In part b only the horizontal velocity component v sub x is shown whose magnitude is constant at various positions in the path. In part c only vertical velocity component v sub y is shown. The vertical velocity component v sub y is upwards till it reaches the maximum point and then its direction changes to downwards. In part d resultant v of horizontal velocity component v sub x and downward vertical velocity component v sub y is found which makes an angle theta with the horizontal x axis. The direction of resultant velocity v is towards south east.\" height=\"600\"><\/span><\/p><\/div>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id2175010\">\n<div data-type=\"title\" class=\"title\">A Fireworks Projectile Explodes High and Away<\/div>\n<p id=\"import-auto-id1896064\">During a fireworks display, a shell is shot into the air with an initial speed of 70.0 m\/s at an angle of [latex]75.0\u00ba[\/latex] above the horizontal, as illustrated in <a href=\"#import-auto-id934168\" class=\"autogenerated-content\">(Figure)<\/a>. The fuse is timed to ignite the shell just as it reaches its highest point above the ground. (a) Calculate the height at which the shell explodes. (b) How much time passed between the launch of the shell and the explosion? (c) What is the horizontal displacement of the shell when it explodes?<\/p>\n<p id=\"eip-149\"><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id1629969\">Because air resistance is negligible for the unexploded shell, the analysis method outlined above can be used. The motion can be broken into horizontal and vertical motions in which  [latex]{a}_{x}=0[\/latex] and  <\/p>\n<p>[latex]{a}_{y}=\u2013g[\/latex]. We can then define <\/p>\n<p>[latex]{x}_{0}[\/latex] and [latex]{y}_{0}[\/latex] to be zero and solve for the desired quantities.<\/p>\n<p id=\"eip-774\"><strong>Solution for (a)<\/strong><\/p>\n<p id=\"import-auto-id1669571\">By \u201cheight\u201d we mean the altitude or vertical position [latex]y[\/latex] above the starting point. The highest point in any trajectory, called the apex, is reached when [latex]{v}_{y}=0[\/latex]. Since we know the initial and final velocities as well as the initial position, we use the following equation to find [latex]y[\/latex]:<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-734\">[latex]{v}_{y}^{2}={v}_{0y}^{2}-2g\\left(y-{y}_{0}\\right)\\text{.}[\/latex]<\/div>\n<div class=\"bc-figure figure\" id=\"import-auto-id934168\">\n<div class=\"bc-figcaption figcaption\">The trajectory of a fireworks shell. The fuse is set to explode the shell at the highest point in its trajectory, which is found to be at a height of 233 m and 125 m away horizontally.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1576299\" data-alt=\"The x y graph shows the trajectory of fireworks shell. The initial velocity of the shell v zero is at angle theta zero equal to seventy five degrees with the horizontal x axis. The fuse is set to explode the shell at the highest point of the trajectory which is at a height h equal to two hundred thirty three meters and at a horizontal distance x equal to one hundred twenty five meters from the origin.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_04_03a.jpg\" data-media-type=\"image\/jpg\" alt=\"The x y graph shows the trajectory of fireworks shell. The initial velocity of the shell v zero is at angle theta zero equal to seventy five degrees with the horizontal x axis. The fuse is set to explode the shell at the highest point of the trajectory which is at a height h equal to two hundred thirty three meters and at a horizontal distance x equal to one hundred twenty five meters from the origin.\" height=\"250\"><\/span><\/p><\/div>\n<p id=\"import-auto-id1163607\">Because [latex]{y}_{0}[\/latex] and [latex]{v}_{y}[\/latex] are both zero, the equation simplifies to<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]0={v}_{0y}^{2}-2\\text{gy.}[\/latex]<\/div>\n<p id=\"import-auto-id2114969\">Solving for [latex]y[\/latex] gives<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-256\">[latex]y=\\frac{{v}_{0y}^{2}}{2g}\\text{.}[\/latex]<\/div>\n<p id=\"import-auto-id1877237\">Now we must find <\/p>\n<p>[latex]{v}_{0y}[\/latex], the component of the initial velocity in the <em data-effect=\"italics\">y<\/em>-direction. It is given by <\/p>\n<p>[latex]{v}_{0y}={v}_{{0}^{}}\\phantom{\\rule{0.25em}{0ex}}\\text{sin}\\phantom{\\rule{0.25em}{0ex}}\\theta [\/latex], where <\/p>\n<p>[latex]{v}_{0y}[\/latex] is the initial velocity of 70.0 m\/s, and <\/p>\n<p>[latex]{\\theta }_{0}=75.0\u00ba[\/latex] is the initial angle. Thus,<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-677\">[latex]{v}_{0y}={v}_{0}\\phantom{\\rule{0.25em}{0ex}}\\text{sin}\\phantom{\\rule{0.25em}{0ex}}{\\theta }_{0}=\\left(\\text{70.0 m\/s}\\right)\\left(\\text{sin 75\u00ba}\\right)=\\text{67.6 m\/s.}[\/latex]<\/div>\n<p id=\"import-auto-id2271493\">and [latex]y[\/latex] is<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-512\">[latex]y=\\frac{\\left(\\text{67}\\text{.6 m\/s}{\\right)}^{2}}{2\\left(9\\text{.}\\text{80 m}{\\text{\/s}}^{2}\\right)},[\/latex]<\/div>\n<p id=\"import-auto-id1919502\">so that<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-310\">[latex]y=\\text{233}\\text{m.}[\/latex]<\/div>\n<p id=\"eip-429\"><strong>Discussion for (a)<\/strong><\/p>\n<p id=\"import-auto-id2165781\">Note that because up is positive, the initial velocity is positive, as is the maximum height, but the acceleration due to gravity is negative. Note also that the maximum height depends only on the vertical component of the initial velocity, so that any projectile with a 67.6 m\/s initial vertical component of velocity will reach a maximum height of 233 m (neglecting air resistance). The numbers in this example are reasonable for large fireworks displays, the shells of which do reach such heights before exploding. In practice, air resistance is not completely negligible, and so the initial velocity would have to be somewhat larger than that given to reach the same height.<\/p>\n<p id=\"eip-449\"><strong>Solution for (b)<\/strong><\/p>\n<p id=\"import-auto-id1632028\">As in many physics problems, there is more than one way to solve for the time to the highest point. In this case, the easiest method is to use [latex]y={y}_{0}+\\frac{1}{2}\\left({v}_{0y}+{v}_{y}\\right)t[\/latex]. Because [latex]{y}_{0}[\/latex] is zero, this equation reduces to simply<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-383\">[latex]y=\\frac{1}{2}\\left({v}_{0y}+{v}_{y}\\right)t\\text{.}[\/latex]<\/div>\n<p id=\"import-auto-id1383088\">Note that the final vertical velocity, [latex]{v}_{y}[\/latex], at the highest point is zero. Thus,<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-50\">[latex]\\begin{array}{lll}t&amp; =&amp; \\frac{2y}{\\left({v}_{0y}+{v}_{y}\\right)}=\\frac{2\\left(\\text{233 m}\\right)}{\\left(\\text{67.6 m\/s}\\right)}\\\\ &amp; =&amp; \\text{6.90 s}\\text{.}\\end{array}[\/latex]<\/div>\n<p id=\"eip-31\"><strong>Discussion for (b)<\/strong><\/p>\n<p id=\"import-auto-id1848626\">This time is also reasonable for large fireworks. When you are able to see the launch of fireworks, you will notice several seconds pass before the shell explodes. (Another way of finding the time is by using [latex]y={y}_{0}+{v}_{0y}t-\\frac{1}{2}{\\text{gt}}^{2}[\/latex], and solving the quadratic equation for [latex]t[\/latex].)<\/p>\n<p id=\"eip-939\"><strong>Solution for (c)<\/strong><\/p>\n<p id=\"import-auto-id2262600\">Because air resistance is negligible, [latex]{a}_{x}=0[\/latex] and the horizontal velocity is constant, as discussed above. The horizontal displacement is horizontal velocity multiplied by time as given by [latex]x={x}_{0}+{v}_{x}t[\/latex], where [latex]{x}_{0}[\/latex] is equal to zero:<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-675\">[latex]x={v}_{x}t\\text{,}[\/latex]<\/div>\n<p id=\"import-auto-id1871833\">where [latex]{v}_{x}[\/latex] is the <em data-effect=\"italics\"><em data-effect=\"italics\">x<\/em><\/em>-component of the velocity, which is given by [latex]{v}_{x}={v}_{0}\\phantom{\\rule{0.25em}{0ex}}\\text{cos}\\phantom{\\rule{0.25em}{0ex}}{\\theta }_{0}\\text{.}[\/latex] Now,<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-884\">[latex]{v}_{x}={v}_{0}\\phantom{\\rule{0.25em}{0ex}}\\text{cos}\\phantom{\\rule{0.25em}{0ex}}{\\theta }_{0}=\\left(\\text{70}\\text{.}0 m\/s\\text{}\\right)\\left(\\text{cos 75.0\u00ba}\\right)=\\text{18}\\text{.}1 m\/s.[\/latex]<\/div>\n<p id=\"import-auto-id2046887\">The time [latex]t[\/latex] for both motions is the same, and so [latex]x[\/latex] is<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-685\">[latex]x=\\left(\\text{18}\\text{.}1 m\/s\\text{}\\right)\\left(6\\text{.}\\text{90 s}\\text{}\\right)=\\text{125 m.}\\text{}[\/latex]<\/div>\n<p id=\"eip-247\"><strong>Discussion for (c)<\/strong><\/p>\n<p id=\"import-auto-id1857186\">The horizontal motion is a constant velocity in the absence of air resistance. The horizontal displacement found here could be useful in keeping the fireworks fragments from falling on spectators. Once the shell explodes, air resistance has a major effect, and many fragments will land directly below.<\/p>\n<\/div>\n<p id=\"import-auto-id1986266\">In solving part (a) of the preceding example, the expression we found for [latex]y[\/latex] is valid for any projectile motion where air resistance is negligible. Call the maximum height [latex]y=h[\/latex]; then,<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-803\">[latex]h=\\frac{{v}_{0y}^{2}}{2g}\\text{.}[\/latex]<\/div>\n<p id=\"import-auto-id1973673\">This equation defines the <em data-effect=\"italics\"><em data-effect=\"italics\">maximum height of a projectile<\/em><\/em> and depends only on the vertical component of the initial velocity.<\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id1479427\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Defining a Coordinate System<\/div>\n<p id=\"import-auto-id2275341\">It is important to set up a coordinate system when analyzing projectile motion. One part of defining the coordinate system is to define an origin for the [latex]x[\/latex] and [latex]y[\/latex] positions. Often, it is convenient to choose the initial position of the object as the origin such that [latex]{x}_{0}=0[\/latex] and [latex]{y}_{0}=0[\/latex]. It is also important to define the positive and negative directions in the [latex]x[\/latex] and [latex]y[\/latex] directions. Typically, we define the positive vertical direction as upwards, and the positive horizontal direction is usually the direction of the object\u2019s motion. When this is the case, the vertical acceleration, [latex]g[\/latex], takes a negative value (since it is directed downwards towards the Earth). However, it is occasionally useful to define the coordinates differently. For example, if you are analyzing the motion of a ball thrown downwards from the top of a cliff, it may make sense to define the positive direction downwards since the motion of the ball is solely in the downwards direction. If this is the case, [latex]g[\/latex] takes a positive value.<\/p>\n<\/div>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id708626\">\n<div data-type=\"title\" class=\"title\">Calculating Projectile Motion: Hot Rock Projectile<\/div>\n<p id=\"import-auto-id1916950\">Kilauea in Hawaii is the world\u2019s most continuously active volcano. Very active volcanoes characteristically eject red-hot rocks and lava rather than smoke and ash. Suppose a large rock is ejected from the volcano with a speed of 25.0 m\/s and at an angle [latex]\\text{35.0\u00ba}[\/latex] above the horizontal, as shown in <a href=\"#import-auto-id1817519\" class=\"autogenerated-content\">(Figure)<\/a>. The rock strikes the side of the volcano at an altitude 20.0 m lower than its starting point. (a) Calculate the time it takes the rock to follow this path. (b) What are the magnitude and direction of the rock\u2019s velocity at impact?<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1817519\">\n<div class=\"bc-figcaption figcaption\">The trajectory of a rock ejected from the Kilauea volcano.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1817520\" data-alt=\"The trajectory of a rock ejected from a volcano is shown. The initial velocity of rock v zero is equal to twenty five meters per second and it makes an angle of thirty five degrees with the horizontal x axis. The figure shows rock falling down a height of twenty meters below the volcano level. The velocity at this point is v which makes an angle of theta with horizontal x axis. The direction of v is south east.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_04_04a.jpg\" data-media-type=\"image\/jpg\" alt=\"The trajectory of a rock ejected from a volcano is shown. The initial velocity of rock v zero is equal to twenty five meters per second and it makes an angle of thirty five degrees with the horizontal x axis. The figure shows rock falling down a height of twenty meters below the volcano level. The velocity at this point is v which makes an angle of theta with horizontal x axis. The direction of v is south east.\" width=\"400\"><\/span><\/p><\/div>\n<p id=\"eip-770\"><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id1823692\">Again, resolving this two-dimensional motion into two independent one-dimensional motions will allow us to solve for the desired quantities. The time a projectile is in the air is governed by its vertical motion alone. We will solve for [latex]t[\/latex] first. While the rock is rising and falling vertically, the horizontal motion continues at a constant velocity. This example asks for the final velocity. Thus, the vertical and horizontal results will be recombined to obtain [latex]v[\/latex] and [latex]{\\theta }_{v}[\/latex] at the final time [latex]t[\/latex] determined in the first part of the example.<\/p>\n<p id=\"eip-408\"><strong>Solution for (a)<\/strong><\/p>\n<p id=\"import-auto-id1608071\">While the rock is in the air, it rises and then falls to a final position 20.0 m lower than its starting altitude. We can find the time for this by using<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-895\">[latex]y={y}_{0}+{v}_{0y}t-\\frac{1}{2}{\\text{gt}}^{2}\\text{.}[\/latex]<\/div>\n<p id=\"import-auto-id1843834\">If we take the initial position [latex]{y}_{0}[\/latex] to be zero, then the final position is [latex]y=-\\text{20}\\text{.0 m}\\text{.}[\/latex] Now the initial vertical velocity is the vertical component of the initial velocity, found from  [latex]{v}_{0y}={v}_{0}\\phantom{\\rule{0.25em}{0ex}}\\text{sin}\\phantom{\\rule{0.25em}{0ex}}{\\theta }_{0}[\/latex] = ([latex]\\text{25}\\text{.}\\text{0&nbsp;m\/s}[\/latex])([latex]\\text{sin 35.0\u00ba}[\/latex]) = [latex]\\text{14}\\text{.}\\text{3&nbsp;m\/s}[\/latex]. Substituting known values yields<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]-\\text{20}\\text{.}0 m\\text{}=\\left(\\text{14}\\text{.}3 m\/s\\text{}\\right)t-\\left(4\\text{.}\\text{90 m\/s}{\\text{}}^{2}\\right){t}^{2}\\text{.}[\/latex]<\/div>\n<p id=\"import-auto-id1561988\">Rearranging terms gives a quadratic equation in [latex]t[\/latex]:<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-931\">[latex]\\left(4\\text{.}\\text{90 m\/s}{\\text{}}^{2}\\right){t}^{2}-\\left(\\text{14}\\text{.}\\text{3 m\/s}\\right)t-\\left(\\text{20.0 m}\\right)=0.[\/latex]<\/div>\n<p id=\"import-auto-id2244917\">This expression is a quadratic equation of the form<br>\n[latex]{\\mathrm{at}}^{2}+\\mathrm{bt}+c=0[\/latex], where the constants are <\/p>\n<p>[latex]a=4.90[\/latex], <\/p>\n<p>[latex]b=\u201314.3[\/latex], and <\/p>\n<p>[latex]c=\u201320.0.[\/latex] Its solutions are given by the quadratic formula:<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-880\">[latex]t=\\frac{-b\u00b1\\sqrt{{b}^{2}-4\\text{ac}}}{\\text{2}\\text{a}}\\text{.}[\/latex]<\/div>\n<p id=\"import-auto-id1600199\">This equation yields two solutions: <\/p>\n<p>[latex]t=3.96[\/latex] and <\/p>\n<p>[latex]t=\u20131.03[\/latex]. (It is left as an exercise for the reader to verify these solutions.) The time is <\/p>\n<p>[latex]t=3.96\\phantom{\\rule{0.25em}{0ex}}\\text{s}[\/latex] or <\/p>\n<p>[latex]\u20131.03\\phantom{\\rule{0.25em}{0ex}}\\text{s}[\/latex]. The negative value of time implies an event before the start of motion, and so we discard it. Thus,<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-267\">[latex]t=3\\text{.}\\text{96 s}\\text{.}[\/latex]<\/div>\n<p id=\"eip-46\"><strong>Discussion for (a)<\/strong><\/p>\n<p id=\"import-auto-id1368589\">The time for projectile motion is completely determined by the vertical motion. So any projectile that has an initial vertical velocity of 14.3 m\/s and lands 20.0 m below its starting altitude will spend 3.96 s in the air.<\/p>\n<p id=\"eip-653\"><strong>Solution for (b)<\/strong><\/p>\n<p id=\"import-auto-id1599448\">From the information now in hand, we can find the final horizontal and vertical velocities [latex]{v}_{x}[\/latex] and [latex]{v}_{y}[\/latex] and combine them to find the total velocity [latex]v[\/latex] and the angle [latex]{\\theta }_{0}[\/latex] it makes with the horizontal. Of course, [latex]{v}_{x}[\/latex] is constant so we can solve for it at any horizontal location. In this case, we chose the starting point since we know both the initial velocity and initial angle. Therefore:<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-873\">[latex]{v}_{x}={v}_{0}\\phantom{\\rule{0.25em}{0ex}}\\text{cos}\\phantom{\\rule{0.25em}{0ex}}{\\theta }_{0}=\\left(\\text{25}\\text{.}0 m\/s\\text{}\\right)\\left(\\text{cos 35\u00ba}\\right)=\\text{20}\\text{.}5 m\/s.\\text{}[\/latex]<\/div>\n<p id=\"import-auto-id2252925\">The final vertical velocity is given by the following equation:<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-168\">[latex]{v}_{y}={v}_{0y}-\\text{gt,}[\/latex]<\/div>\n<p id=\"import-auto-id2173689\">where [latex]{v}_{0y}[\/latex] was found in part (a) to be [latex]\\text{14}\\text{.}\\text{3&nbsp;m\/s}[\/latex]. Thus,<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-113\">[latex]{v}_{y}=\\text{14}\\text{.}3 m\/s\\text{}-\\left(9\\text{.}\\text{80 m\/s}{\\text{}}^{2}\\right)\\left(3\\text{.}\\text{96 s}\\text{}\\right)[\/latex]<\/div>\n<p id=\"import-auto-id1792451\">so that<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-571\">[latex]{v}_{y}=-\\text{24}\\text{.}5 m\/s.\\text{}[\/latex]<\/div>\n<p id=\"import-auto-id2239982\">To find the magnitude of the final velocity [latex]v[\/latex] we combine its perpendicular components, using the following equation:<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-394\">[latex]v=\\sqrt{{v}_{x}^{2}+{v}_{y}^{2}}=\\sqrt{\\left(\\text{20}\\text{.}5 m\/s\\text{}{\\right)}^{2}+\\left(-\\text{24}\\text{.}5 m\/s\\text{}{\\right)}^{2}}\\text{,}[\/latex]<\/div>\n<p id=\"import-auto-id1677955\">which gives<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-60\">[latex]v=\\text{31}\\text{.}9 m\/s.\\text{}[\/latex]<\/div>\n<p id=\"import-auto-id1645980\">The direction [latex]{\\theta }_{v}[\/latex] is found from the equation:<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-353\">[latex]{\\theta }_{v}={\\text{tan}}^{-1}\\left({v}_{y}\/{v}_{x}\\right)[\/latex]<\/div>\n<p id=\"import-auto-id1972156\">so that<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-589\">[latex]{\\theta }_{v}={\\text{tan}}^{-1}\\left(-\\text{24}\\text{.}5\/\\text{20}\\text{.}5\\right)={\\text{tan}}^{-1}\\left(-1\\text{.}\\text{19}\\right)\\text{.}[\/latex]<\/div>\n<p id=\"import-auto-id1613163\">Thus,<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]{\\theta }_{v}=-\\text{50}\\text{.}1\u00ba\\text{.}[\/latex]<\/div>\n<p id=\"eip-299\"><strong>Discussion for (b)<\/strong><\/p>\n<p id=\"import-auto-id1459619\">The negative angle means that the velocity is [latex]\\text{50}\\text{.}1\u00ba[\/latex] below the horizontal. This result is consistent with the fact that the final vertical velocity is negative and hence downward\u2014as you would expect because the final altitude is 20.0 m lower than the initial altitude. (See <a href=\"#import-auto-id1817519\" class=\"autogenerated-content\">(Figure)<\/a>.)<\/p>\n<\/div>\n<p id=\"import-auto-id1532818\">One of the most important things illustrated by projectile motion is that vertical and horizontal motions are independent of each other. Galileo was the first person to fully comprehend this characteristic. He used it to predict the range of a projectile. On level ground, we define <span data-type=\"term\" id=\"import-auto-id1751163\">range<\/span> to be the horizontal distance [latex]R[\/latex] traveled by a projectile. Galileo and many others were interested in the range of projectiles primarily for military purposes\u2014such as aiming cannons. However, investigating the range of projectiles can shed light on other interesting phenomena, such as the orbits of satellites around the Earth. Let us consider projectile range further.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1904800\">\n<div class=\"bc-figcaption figcaption\">Trajectories of projectiles on level ground. (a) The greater the initial speed [latex]{v}_{0}[\/latex], the greater the range for a given initial angle. (b) The effect of initial angle [latex]{\\theta }_{0}[\/latex] on the range of a projectile with a given initial speed. Note that the range is the same for [latex]\\text{15\u00ba}[\/latex] and [latex]\\text{75\u00ba}[\/latex], although the maximum heights of those paths are different.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1904802\" data-alt=\"Part a of the figure shows three different trajectories of projectiles on level ground. In each case the projectiles makes an angle of forty five degrees with the horizontal axis. The first projectile of initial velocity thirty meters per second travels a horizontal distance of R equal to ninety one point eight meters. The second projectile of initial velocity forty meters per second travels a horizontal distance of R equal to one hundred sixty three meters. The third projectile of initial velocity fifty meters per second travels a horizontal distance of R equal to two hundred fifty five meters.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_04_05a.jpg\" data-media-type=\"image\/jpg\" alt=\"Part a of the figure shows three different trajectories of projectiles on level ground. In each case the projectiles makes an angle of forty five degrees with the horizontal axis. The first projectile of initial velocity thirty meters per second travels a horizontal distance of R equal to ninety one point eight meters. The second projectile of initial velocity forty meters per second travels a horizontal distance of R equal to one hundred sixty three meters. The third projectile of initial velocity fifty meters per second travels a horizontal distance of R equal to two hundred fifty five meters.\" height=\"300\"><\/span><\/p><\/div>\n<p id=\"import-auto-id1750749\">How does the initial velocity of a projectile affect its range? Obviously, the greater the initial speed [latex]{v}_{0}[\/latex], the greater the range, as shown in <a href=\"#import-auto-id1904800\" class=\"autogenerated-content\">(Figure)<\/a>(a). The initial angle [latex]{\\theta }_{0}[\/latex] also has a dramatic effect on the range, as illustrated in <a href=\"#import-auto-id1904800\" class=\"autogenerated-content\">(Figure)<\/a>(b). For a fixed initial speed, such as might be produced by a cannon, the maximum range is obtained with [latex]{\\theta }_{0}=\\text{45\u00ba}[\/latex]. This is true only for conditions neglecting air resistance. If air resistance is considered, the maximum angle is approximately [latex]\\text{38\u00ba}[\/latex]. Interestingly, for every initial angle except [latex]\\text{45\u00ba}[\/latex], there are two angles that give the same range\u2014the sum of those angles is [latex]\\text{90\u00ba}[\/latex]. The range also depends on the value of the acceleration of gravity [latex]g[\/latex]. The lunar astronaut Alan Shepherd was able to drive a golf ball a great distance on the Moon because gravity is weaker there. The range [latex]R[\/latex] of a projectile on <em data-effect=\"italics\"><em data-effect=\"italics\">level ground<\/em><\/em> for which air resistance is negligible is given by <\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-240\">[latex]R=\\frac{{v}_{0}^{2}\\phantom{\\rule{0.25em}{0ex}}\\text{sin}\\phantom{\\rule{0.25em}{0ex}}{2\\theta }_{0}}{g}\\text{,}[\/latex]<\/div>\n<p id=\"import-auto-id1674836\">where [latex]{v}_{0}[\/latex] is the initial speed and [latex]{\\theta }_{0}[\/latex] is the initial angle relative to the horizontal. The proof of this equation is left as an end-of-chapter problem (hints are given), but it does fit the major features of projectile range as described.<\/p>\n<p id=\"import-auto-id1686986\">When we speak of the range of a projectile on level ground, we assume that [latex]R[\/latex] is very small compared with the circumference of the Earth. If, however, the range is large, the Earth curves away below the projectile and acceleration of gravity changes direction along the path. The range is larger than predicted by the range equation given above because the projectile has farther to fall than it would on level ground. (See <a href=\"#import-auto-id1645881\" class=\"autogenerated-content\">(Figure)<\/a>.) If the initial speed is great enough, the projectile goes into orbit.  This possibility was recognized centuries before it could be accomplished. When an object is in orbit, the Earth curves away from underneath the object at the same rate as it falls. The object thus falls continuously but never hits the surface. These and other aspects of orbital motion, such as the rotation of the Earth, will be covered analytically and in greater depth later in this text.<\/p>\n<p id=\"import-auto-id1645878\">Once again we see that thinking about one topic, such as the range of a projectile, can lead us to others, such as the Earth orbits. In  <a href=\"\/contents\/b64da007-2df2-4425-8f4b-e4d42ed423d9@11\">Addition of Velocities<\/a>, we will examine the addition of velocities, which is another important aspect of two-dimensional kinematics and will also yield insights beyond the immediate topic.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1645881\">\n<div class=\"bc-figcaption figcaption\">Projectile to satellite. In each case shown here, a projectile is launched from a very high tower to avoid air resistance. With increasing initial speed, the range increases and becomes longer than it would be on level ground because the Earth curves away underneath its path. With a large enough initial speed, orbit is achieved.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1825851\" data-alt=\"A figure of the Earth is shown and on top of it a very high tower is placed. A projectile satellite is launched from this very high tower with initial velocity of v zero in the horizontal direction. Several trajectories are shown with increasing range. A circular trajectory is shown indicating the satellite achieved its orbit and it is revolving around the Earth.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_04_06a.jpg\" data-media-type=\"image\/jpg\" alt=\"A figure of the Earth is shown and on top of it a very high tower is placed. A projectile satellite is launched from this very high tower with initial velocity of v zero in the horizontal direction. Several trajectories are shown with increasing range. A circular trajectory is shown indicating the satellite achieved its orbit and it is revolving around the Earth.\" width=\"200\"><\/span><\/p><\/div>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"eip-89\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">PhET Explorations: Projectile Motion<\/div>\n<p id=\"eip-id1169738107387\">Blast a Buick out of a cannon! Learn about projectile motion by firing various objects. Set the angle, initial speed, and mass. Add air resistance. Make a game out of this simulation by trying to hit a target.<\/p>\n<div class=\"bc-figure figure\" id=\"eip-id1462984\">\n<div class=\"bc-figcaption figcaption\"><a href=\"\/resources\/5f60797f22fa74f2a6b84d7fe2c9f149f1109c19\/projectile-motion_en.jar\">Projectile Motion<\/a><\/div>\n<p><span data-type=\"media\" id=\"Phet_module_3.4\" data-alt=\"\"><a href=\"\/resources\/5f60797f22fa74f2a6b84d7fe2c9f149f1109c19\/projectile-motion_en.jar\" data-type=\"image\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/PhET_Icon.png\" data-media-type=\"image\/png\" alt=\"\" data-print=\"false\" width=\"450\"><\/a><span data-media-type=\"image\/png\" data-print=\"true\" data-src=\"\/resources\/075500ad9f71890a85fe3f7a4137ac08e2b7907c\/PhET_Icon.png\" data-type=\"image\"><\/span><\/span><\/p><\/div>\n<\/div>\n<div class=\"section-summary\" data-depth=\"1\" id=\"fs-id1843457\">\n<h1 data-type=\"title\">Summary<\/h1>\n<ul id=\"fs-id2130996\">\n<li id=\"import-auto-id1677012\">Projectile motion is the motion of an object through the air that is subject only to the acceleration of gravity.<\/li>\n<li id=\"import-auto-id1786765\">To solve projectile motion problems, perform the following steps:\n<ol id=\"fs-id1842700\" data-number-style=\"arabic\" class=\"stepwise\">\n<li id=\"import-auto-id1830314\">Determine a coordinate system. Then, resolve the position and\/or velocity of the object in the horizontal and vertical components. The components of position [latex]\\mathbf{s}[\/latex] are given by the quantities [latex]x[\/latex] and [latex]y[\/latex], and the components of the velocity [latex]\\mathbf{v}[\/latex] are given by [latex]{v}_{x}=v\\phantom{\\rule{0.25em}{0ex}}\\text{cos}\\phantom{\\rule{0.25em}{0ex}}\\theta [\/latex] and [latex]{v}_{y}=v\\phantom{\\rule{0.25em}{0ex}}\\text{sin}\\phantom{\\rule{0.25em}{0ex}}\\theta [\/latex], where [latex]v[\/latex] is the magnitude of the velocity and [latex]\\theta [\/latex] is its direction.<\/li>\n<li id=\"import-auto-id1830316\">Analyze the motion of the projectile in the horizontal direction using the following equations:\n<div data-type=\"equation\" class=\"equation\" id=\"eip-898\">[latex]\\text{Horizontal motion}\\left({a}_{x}=0\\right)[\/latex]<\/div>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-236\">[latex]x={x}_{0}+{v}_{x}t[\/latex]<\/div>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-612\">[latex]{v}_{x}={v}_{0x}={\\mathbf{\\text{v}}}_{\\text{x}}=\\text{velocity is a constant.}[\/latex]<\/div>\n<\/li>\n<li id=\"import-auto-id1939082\">Analyze the motion of the projectile in the vertical direction using the following equations:\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id1939084\">[latex]\\text{Vertical motion}\\left(\\text{Assuming positive direction is up;}\\phantom{\\rule{0.25em}{0ex}}{a}_{y}=-g=-9\\text{.}\\text{80 m}{\\text{\/s}}^{2}\\right)[\/latex]<\/div>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id1492830\">[latex]y={y}_{0}+\\frac{1}{2}\\left({v}_{0y}+{v}_{y}\\right)t[\/latex]<\/div>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2022844\">[latex]{v}_{y}={v}_{0y}-\\text{gt}[\/latex]<\/div>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id1677876\">[latex]y={y}_{0}+{v}_{0y}t-\\frac{1}{2}{\\text{gt}}^{2}[\/latex]<\/div>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id1653540\">[latex]{v}_{y}^{2}={v}_{0y}^{2}-2g\\left(y-{y}_{0}\\right).[\/latex]<\/div>\n<\/li>\n<li id=\"import-auto-id1552181\">Recombine the horizontal and vertical components of location and\/or velocity using the following equations:\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2092332\">[latex]s=\\sqrt{{x}^{2}+{y}^{2}}[\/latex]<\/div>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2282348\">[latex]\\theta ={\\text{tan}}^{-1}\\left(y\/x\\right)[\/latex]<\/div>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2274748\">[latex]v=\\sqrt{{v}_{x}^{2}+{v}_{y}^{2}}[\/latex]<\/div>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id1979208\">[latex]{\\theta }_{\\text{v}}={\\text{tan}}^{-1}\\left({v}_{y}\/{v}_{x}\\right).[\/latex]<\/div>\n<\/li>\n<\/ol>\n<\/li>\n<li id=\"import-auto-id1888635\">The maximum height [latex]h[\/latex] of a projectile launched with initial vertical velocity [latex]{v}_{0y}[\/latex] is given by\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id1534227\">[latex]h=\\frac{{v}_{0y}^{2}}{2g}.[\/latex]<\/div>\n<\/li>\n<li id=\"import-auto-id1823593\">The maximum horizontal distance traveled by a projectile is called the <strong>range<\/strong>. The range [latex]R[\/latex] of a projectile on level ground launched at an angle [latex]{\\theta }_{0}[\/latex] above the horizontal with initial speed [latex]{v}_{0}[\/latex] is given by\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id1951750\">[latex]R=\\frac{{v}_{0}^{2}\\phantom{\\rule{0.25em}{0ex}}\\text{sin}\\phantom{\\rule{0.25em}{0ex}}{2\\theta }_{0}}{g}.[\/latex]<\/div>\n<\/li>\n<\/ul>\n<\/div>\n<div class=\"conceptual-questions\" data-depth=\"1\" id=\"fs-id2865659\" data-element-type=\"conceptual-questions\">\n<h1 data-type=\"title\">Conceptual Questions<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2183300\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2183302\">\n<p id=\"fs-id2298558\">Answer the following questions for projectile motion on level ground assuming negligible air resistance (the initial angle being neither [latex]\\text{0\u00ba}[\/latex] nor [latex]\\text{90\u00ba}[\/latex]): (a) Is the velocity ever zero? (b) When is the velocity a minimum? A maximum? (c) Can the velocity ever be the same as the initial velocity at a time other than at [latex]t=0[\/latex]? (d) Can the speed ever be the same as the initial speed at a time other than at [latex]t=0[\/latex]?\n        <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1638420\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1638423\">\n<p id=\"fs-id1638424\">Answer the following questions for projectile motion on level ground assuming negligible air resistance (the initial angle being neither [latex]\\text{0\u00ba}[\/latex] nor [latex]\\text{90\u00ba}[\/latex]): (a) Is the acceleration ever zero? (b) Is the acceleration ever in the same direction as a component of velocity? (c) Is the acceleration ever opposite in direction to a component of velocity?\n        <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2062475\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2062477\">\n<p id=\"fs-id2062478\">For a fixed initial speed, the range of a projectile is determined by the angle at which it is fired. For all but the maximum, there are two angles that give the same range. Considering factors that might affect the ability of an archer to hit a target, such as wind, explain why the smaller angle (closer to the horizontal) is preferable. When would it be necessary for the archer to use the larger angle? Why does the punter in a football game use the higher trajectory?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1875651\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1875652\">\n<p id=\"fs-id1875653\">During a lecture demonstration, a professor places two coins on the edge of a table. She then flicks one of the coins horizontally off the table, simultaneously nudging the other over the edge. Describe the subsequent motion of the two coins, in particular discussing whether they hit the floor at the same time.\n        <\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"problems-exercises\" data-depth=\"1\" id=\"fs-id1875655\" data-element-type=\"problems-exercises\">\n<h1 data-type=\"title\">Problems &amp; Exercises<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1923898\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1105650\">\n<p id=\"fs-id1105652\">A projectile is launched at ground level with an initial speed of 50.0 m\/s at an angle of [latex]30.0\u00ba[\/latex] above the horizontal. It strikes a target above the ground 3.00 seconds later. What are the [latex]x[\/latex] and [latex]y[\/latex] distances from where the projectile was launched to where it lands?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1543587\">\n<p id=\"fs-id1915206\">[latex]\\begin{array}{lll}x&amp; =&amp; \\text{1.30 m}\u00d7{10}^{2}\\\\ y&amp; =&amp; \\text{30}\\text{.9 m.}\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1275043\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1019417\">\n<p id=\"fs-id1980036\">A ball is kicked with an initial velocity of 16 m\/s in the horizontal direction and 12 m\/s in the vertical direction. (a) At what speed does the ball hit the ground? (b) For how long does the ball remain in the air? (c)What maximum height is attained by the ball?\n      <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2889503\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1790979\">\n<p id=\"fs-id1790980\">A ball is thrown horizontally from the top of a 60.0-m building and lands 100.0 m from the base of the building. Ignore air resistance. (a) How long is the ball in the air? (b) What must have been the initial horizontal component of the velocity? (c) What is the vertical component of the velocity just before the ball hits the ground? (d) What is the velocity (including both the horizontal and vertical components) of the ball just before it hits the ground?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1945253\">\n<p id=\"fs-id2167208\">(a) 3.50 s<\/p>\n<p id=\"fs-id1612943\">(b) 28.6  m\/s (c) 34.3 m\/s<\/p>\n<p id=\"fs-id916496\">(d) 44.7 m\/s,  [latex]50.2\u00ba[\/latex] below horizontal<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2197387\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2261404\">\n<p id=\"fs-id2261405\">(a) A daredevil is attempting to jump his motorcycle over a line of buses parked end to end by driving up a [latex]\\text{32\u00ba}[\/latex] ramp at a speed of [latex]\\text{40}\\text{.}\\text{0&nbsp;m\/s&nbsp;}\\left(\\text{144&nbsp;km\/h}\\right)[\/latex]. How many buses can he clear if the top of the takeoff ramp is at the same height as the bus tops and the buses are 20.0 m long? (b) Discuss what your answer implies about the margin of error in this act\u2014that is, consider how much greater the range is than the horizontal distance he must travel to miss the end of the last bus. (Neglect air resistance.)\n      <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1420192\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1078714\">\n<p id=\"fs-id1078715\">An archer shoots an arrow at a 75.0 m distant target; the bull\u2019s-eye of the target is at same height as the release height of the arrow. (a) At what angle must the arrow be released to hit the bull\u2019s-eye if its initial speed is 35.0 m\/s? In this part of the problem, explicitly show how you follow the steps involved in solving projectile motion problems. (b) There is a large tree halfway between the archer and the target with an overhanging horizontal branch 3.50 m above the release height of the arrow. Will the arrow go over or under the branch?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1970452\">\n<p id=\"fs-id1970455\">(a) [latex]\\text{18}\\text{.}\\text{4\u00ba}[\/latex]<\/p>\n<p id=\"fs-id1678230\">(b) The arrow will go over the branch.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1934878\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2226553\">\n<p id=\"fs-id2226554\">A rugby player passes the ball 7.00 m across the field, where it is caught at the same height as it left his hand. (a) At what angle was the ball thrown if its initial speed was 12.0 m\/s, assuming that the smaller of the two possible angles was used? (b) What other angle gives the same range, and why would it not be used? (c) How long did this pass take?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2126267\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2889922\">\n<p id=\"fs-id2889923\">Verify the ranges for the projectiles in <a href=\"#import-auto-id1904800\" class=\"autogenerated-content\">(Figure)<\/a>(a) for [latex]\\theta =\\text{45\u00ba}[\/latex] and the given initial velocities.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1903883\">\n<p id=\"fs-id1912801\">[latex]\\begin{array}{}R=\\frac{{{v}_{0}}^{}}{\\text{sin}{2\\theta }_{0}g}\\\\ \\text{For}\\phantom{\\rule{0.25em}{0ex}}\\theta =\\text{45\u00ba},R=\\frac{{{v}_{0}}^{}}{g}\\end{array}[\/latex]<\/p>\n<p id=\"fs-id1630295\">[latex]R=91.8\\phantom{\\rule{0.25em}{0ex}}\\text{m}[\/latex] for <\/p>\n<p>[latex]{v}_{0}=30\\phantom{\\rule{0.25em}{0ex}}\\text{m\/s}[\/latex]; <\/p>\n[latex]R=163\\phantom{\\rule{0.25em}{0ex}}\\text{m}[\/latex]\n<p> for<br>\n[latex]{v}_{0}=40\\phantom{\\rule{0.25em}{0ex}}\\text{m\/s}[\/latex]; <\/p>\n<p>[latex]R=255\\phantom{\\rule{0.25em}{0ex}}\\text{m}[\/latex] for <\/p>\n<p>[latex]{v}_{0}=50\\phantom{\\rule{0.25em}{0ex}}\\text{m\/s}[\/latex].<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2214647\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2214182\">\n<p id=\"fs-id2214183\">Verify the ranges shown for the projectiles in <a href=\"#import-auto-id1904800\" class=\"autogenerated-content\">(Figure)<\/a>(b) for an initial velocity of 50 m\/s at the given initial angles.\n      <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2905201\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1851487\">\n<p id=\"fs-id1851488\">The cannon on a battleship can fire a shell a maximum distance of 32.0 km. (a) Calculate the initial velocity of the shell. (b) What maximum height does it reach? (At its highest, the shell is above 60% of the atmosphere\u2014but air resistance is not really negligible as assumed to make this problem easier.) (c) The ocean is not flat, because the Earth is curved. Assume that the radius of the Earth is [latex]6\\text{.}\\text{37}\u00d7{\\text{10}}^{3}\\phantom{\\rule{0.25em}{0ex}}\\text{km}[\/latex]. How many meters lower will its surface be 32.0 km from the ship along a horizontal line parallel to the surface at the ship? Does your answer imply that error introduced by the assumption of a flat Earth in projectile motion is significant here?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2075683\">\n<p id=\"fs-id1862278\">(a) 560 m\/s<\/p>\n<p id=\"fs-id2252609\">(b) [latex]8\\text{.}\\text{00}\u00d7{\\text{10}}^{3}\\phantom{\\rule{0.25em}{0ex}}\\text{m}[\/latex]<\/p>\n<p id=\"fs-id2262837\">(c) 80.0 m. This error is not significant because it is only 1% of the answer in part (b).<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1925728\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2282381\">\n<p id=\"fs-id2282382\">An arrow is shot from a height of 1.5 m toward a cliff of height [latex]H[\/latex]. It is shot with a velocity of 30 m\/s at an angle of [latex]\\text{60\u00ba}[\/latex] above the horizontal. It lands on the top edge of the cliff 4.0 s later. (a) What is the height of the cliff? (b) What is the maximum height reached by the arrow along its trajectory? (c) What is the arrow\u2019s impact speed just before hitting the cliff?\n      <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1745072\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2275266\">\n<p id=\"fs-id2275267\">In the standing broad jump, one squats and then pushes off with the legs to see how far one can jump. Suppose the extension of the legs from the crouch position is 0.600 m and the acceleration achieved from this position is 1.25 times the acceleration due to gravity, [latex]g[\/latex]. How far can they jump? State your assumptions. (Increased range can be achieved by swinging the arms in the direction of the jump.)<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2864553\">\n<p id=\"fs-id2864555\">1.50 m, assuming launch angle of  [latex]45\u00ba[\/latex]<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1875777\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2864558\">\n<p id=\"fs-id2864559\">The world long jump record is 8.95 m (Mike Powell, USA, 1991). Treated as a projectile, what is the maximum range obtainable by a person if he has a take-off speed of 9.5 m\/s? State your assumptions.\n      <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2254986\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1543443\">\n<p id=\"fs-id1543444\">Serving at a speed of 170 km\/h, a tennis player hits the ball at a height of 2.5 m and an angle [latex]\\theta [\/latex] below the horizontal. The service line is 11.9 m from the net, which is 0.91 m high. What is the angle [latex]\\theta [\/latex] such that the ball just crosses the net? Will the ball land in the service box, whose out line is 6.40 m from the net?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id722706\">\n<p id=\"fs-id2260869\">[latex]\\theta =6.1\u00ba[\/latex]<\/p>\n<p id=\"fs-id2088346\">yes, the ball lands at 5.3 m from the net<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2173828\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2088349\">\n<p id=\"fs-id2088350\">A football quarterback is moving straight backward at a speed of 2.00 m\/s when he throws a pass to a player 18.0 m straight downfield. (a) If the ball is thrown at an angle of [latex]\\text{25\u00ba}[\/latex] relative to the ground and is caught at the same height as it is released, what is its initial speed relative to the ground? (b) How long does it take to get to the receiver? (c) What is its maximum height above its point of release?\n      <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1796436\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1979282\">\n<p id=\"fs-id1979284\">Gun sights are adjusted to aim high to compensate for the effect of gravity, effectively making the gun accurate only for a specific range. (a) If a gun is sighted to hit targets that are at the same height as the gun and 100.0 m away, how low will the bullet hit if aimed directly at a target 150.0 m away? The muzzle velocity of the bullet is 275 m\/s. (b) Discuss qualitatively how a larger muzzle velocity would affect this problem and what would be the effect of air resistance.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2175653\">\n<p id=\"fs-id2175655\">(a) \u22120.486 m<\/p>\n<p id=\"fs-id1461088\">(b) The larger the muzzle velocity, the smaller the deviation in the vertical direction, because the time of flight would be smaller. Air resistance would have the effect of decreasing the time of flight, therefore increasing the vertical deviation.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2177814\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1781566\">\n<p id=\"fs-id1781567\">An eagle is flying horizontally at a speed of 3.00 m\/s when the fish in her talons wiggles loose and falls into the lake 5.00 m below. Calculate the velocity of the fish relative to the water when it hits the water.\n      <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1914025\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2057849\">\n<p id=\"fs-id2057850\">An owl is carrying a mouse to the chicks in its nest. Its position at that time is 4.00 m west and 12.0 m above the center of the 30.0 cm diameter nest. The owl is flying east at 3.50 m\/s at an angle [latex]30.0\u00ba[\/latex] below the horizontal when it accidentally drops the mouse. Is the owl lucky enough to have the mouse hit the nest? To answer this question, calculate the horizontal position of the mouse when it has fallen 12.0 m.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2042074\">\n<p id=\"fs-id2042076\">4.23 m. No, the owl is not lucky; he misses the nest.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1403577\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2865734\">\n<p id=\"fs-id2865735\">Suppose a soccer player kicks the ball from a distance 30 m toward the goal. Find the initial speed of the ball if it just passes over the goal, 2.4 m above the ground, given the initial direction to be [latex]\\text{40\u00ba}[\/latex] above the horizontal.\n      <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2260735\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1789803\">\n<p id=\"fs-id1789804\">Can a goalkeeper at her\/ his goal kick a soccer ball into the opponent\u2019s goal without the ball touching the ground? The distance will be about 95 m. A goalkeeper can give the ball a speed of 30 m\/s.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1985242\">\n<p id=\"fs-id1985244\">No, the maximum range (neglecting air resistance) is about 92 m. <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1437858\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1891285\">\n<p id=\"fs-id1891286\">The free throw line in basketball is 4.57 m (15 ft) from the basket, which is 3.05 m (10 ft) above the floor. A player standing on the free throw line throws the ball with an initial speed of 7.15 m\/s, releasing it at a height of 2.44 m (8 ft) above the floor. At what angle above the horizontal must the ball be thrown to exactly hit the basket? Note that most players will use a large initial angle rather than a flat shot because it allows for a larger margin of error. Explicitly show how you follow the steps involved in solving projectile motion problems.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1827481\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1750926\">\n<p id=\"fs-id1750928\">In 2007, Michael Carter (U.S.) set a world record in the shot put with a throw of 24.77 m. What was the initial speed of the shot if he released it at a height of 2.10 m and threw it at an angle of [latex]38.0\u00ba[\/latex] above the horizontal? (Although the maximum distance for a projectile on level ground is achieved at [latex]\\text{45\u00ba}[\/latex] when air resistance is neglected, the actual angle to achieve maximum range is smaller; thus, [latex]\\text{38\u00ba}[\/latex] will give a longer range than [latex]\\text{45\u00ba}[\/latex] in the shot put.)<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1630701\">\n<p id=\"fs-id2261978\">15.0 m\/s<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1670278\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1945400\">\n<p id=\"fs-id1945401\">A basketball player is running at [latex]5\\text{.}\\text{00&nbsp;m\/s}[\/latex] directly toward the basket when he jumps into the air to dunk the ball. He maintains his horizontal velocity. (a) What vertical velocity does he need to rise 0.750 m above the floor? (b) How far from the basket (measured in the horizontal direction) must he start his jump to reach his maximum height at the same time as he reaches the basket?\n          <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1779635\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1637691\">\n<p id=\"fs-id1637692\">A football player punts the ball at a [latex]45.0\u00ba[\/latex] angle. Without an effect from the wind, the ball would travel 60.0 m horizontally. (a) What is the initial speed of the ball? (b) When the ball is near its maximum height it experiences a brief gust of wind that reduces its horizontal velocity by 1.50 m\/s. What distance does the ball travel horizontally?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1545352\">\n<p id=\"fs-id1545354\">(a) 24.2 m\/s\n<\/p>\n<p id=\"fs-id1796132\">(b) The ball travels a total of 57.4 m with the brief gust of wind.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2046931\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2241558\">\n<p id=\"fs-id2241559\">Prove that the trajectory of a projectile is parabolic, having the form [latex]y=\\text{ax}+{\\text{bx}}^{2}[\/latex]. To obtain this expression, solve the equation <\/p>\n<p>[latex]x={v}_{0x}t[\/latex] for <\/p>\n<p>[latex]t[\/latex] and substitute it into the expression for <\/p>\n<p>[latex]y={v}_{0y}t\u2013\\left(1\/2\\right){\\text{gt}}^{2}[\/latex] (These equations describe the [latex]x[\/latex] and [latex]y[\/latex] positions of a projectile that starts at the origin.) You should obtain an equation of the form [latex]y=\\text{ax}+{\\text{bx}}^{2}[\/latex] where [latex]a[\/latex] and [latex]b[\/latex] are constants.\n<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2133758\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2890360\">\n<p id=\"fs-id2890361\">Derive [latex]R=\\frac{{v}_{0}^{2}\\phantom{\\rule{0.25em}{0ex}}\\text{sin}\\phantom{\\rule{0.25em}{0ex}}{2\\theta }_{0}}{g}[\/latex] for the range of a projectile on level ground by finding the time [latex]t[\/latex] at which [latex]y[\/latex] becomes zero and substituting this value of [latex]t[\/latex] into the expression for [latex]x-{x}_{0}[\/latex], noting that [latex]R=x-{x}_{0}[\/latex]<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"eip-id2932521\">\n<p id=\"eip-id2932523\">[latex]y-{y}_{0}=0={v}_{0y}t-\\frac{1}{2}{\\mathrm{gt}}^{2}=\\left({v}_{0}\\phantom{\\rule{0.25em}{0ex}}\\text{sin}\\phantom{\\rule{0.25em}{0ex}}\\theta \\right)t-\\frac{1}{2}{\\mathrm{gt}}^{2}[\/latex],<\/p>\n<p id=\"eip-id1990384\">so that [latex]t=\\frac{2\\left({v}_{0}\\phantom{\\rule{0.25em}{0ex}}\\text{sin}\\phantom{\\rule{0.25em}{0ex}}\\theta \\right)}{g}[\/latex]<\/p>\n<p id=\"eip-id1355828\">[latex]x-{x}_{0}={v}_{0x}t=\\left({v}_{0}\\phantom{\\rule{0.25em}{0ex}}\\text{cos}\\phantom{\\rule{0.25em}{0ex}}\\theta \\right)t=R,[\/latex] and substituting for [latex]t[\/latex] gives:<\/p>\n<p id=\"eip-id3306953\">[latex]R={v}_{0}\\phantom{\\rule{0.25em}{0ex}}\\text{cos}\\phantom{\\rule{0.25em}{0ex}}\\theta \\left(\\frac{{2v}_{0}\\phantom{\\rule{0.25em}{0ex}}\\text{sin}\\phantom{\\rule{0.25em}{0ex}}\\theta }{g}\\right)=\\frac{{2v}_{0}^{2}\\phantom{\\rule{0.25em}{0ex}}\\text{sin}\\phantom{\\rule{0.25em}{0ex}}\\theta \\phantom{\\rule{0.25em}{0ex}}\\text{cos}\\phantom{\\rule{0.25em}{0ex}}\\theta }{g}[\/latex]<\/p>\n<p id=\"eip-id1599732\">since [latex]2\\phantom{\\rule{0.25em}{0ex}}\\text{sin}\\phantom{\\rule{0.25em}{0ex}}\\theta \\phantom{\\rule{0.25em}{0ex}}\\text{cos}\\phantom{\\rule{0.25em}{0ex}}\\theta =\\text{sin}\\phantom{\\rule{0.25em}{0ex}}2\\theta ,[\/latex] the range is:<\/p>\n<p id=\"eip-id3385487\">[latex]R=\\frac{{{v}_{0}}^{2}\\phantom{\\rule{0.25em}{0ex}}\\text{sin}\\phantom{\\rule{0.25em}{0ex}}2\\theta }{g}[\/latex].<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1794949\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1626931\">\n<p id=\"fs-id1626932\"><strong>Unreasonable Results<\/strong> (a) Find the maximum range of a super cannon that has a muzzle velocity of 4.0 km\/s. (b) What is unreasonable about the range you found? (c) Is the premise unreasonable or is the available equation inapplicable? Explain your answer. (d) If such a muzzle velocity could be obtained, discuss the effects of air resistance, thinning air with altitude, and the curvature of the Earth on the range of the super cannon.\n<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1815382\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2255165\">\n<p id=\"fs-id2255166\"><strong>Construct Your Own Problem<\/strong> Consider a ball tossed over a fence. Construct a problem in which you calculate the ball\u2019s needed initial velocity to just clear the fence. Among the things to determine are; the height of the fence, the distance to the fence from the point of release of the ball, and the height at which the ball is released. You should also consider whether it is possible to choose the initial speed for the ball and just calculate the angle at which it is thrown. Also examine the possibility of multiple solutions given the distances and heights you have chosen.\n <\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"glossary\" class=\"textbox shaded\">\n<h2 data-type=\"glossary-title\">Glossary<\/h2>\n<dl class=\"definition\" id=\"import-auto-id2275857\">\n<dt>air resistance<\/dt>\n<dd id=\"fs-id1668212\">a frictional force that slows the motion of objects as they travel through the air; when solving basic physics problems, air resistance is assumed to be zero<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"fs-id1405173\">\n<dt>kinematics<\/dt>\n<dd id=\"fs-id1949378\">the study of motion without regard to mass or force<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"fs-id1949381\">\n<dt>motion<\/dt>\n<dd id=\"fs-id2324535\">displacement of an object as a function of time<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id1459204\">\n<dt>projectile<\/dt>\n<dd id=\"fs-id1798529\">an object that travels through the air and experiences only acceleration due to gravity<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id1981500\">\n<dt>projectile motion<\/dt>\n<dd id=\"fs-id2253704\">the motion of an object that is subject only to the acceleration of gravity<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id2257218\">\n<dt>range<\/dt>\n<dd id=\"fs-id1883200\">the maximum horizontal distance that a projectile travels<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id2852095\">\n<dt>trajectory<\/dt>\n<dd id=\"fs-id2222621\">the path of a projectile through the air<\/dd>\n<\/dl>\n<\/div>\n\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<ul>\n<li>Identify and explain the properties of a projectile, such as acceleration due to gravity, range, maximum height, and trajectory.<\/li>\n<li>Determine the location and velocity of a projectile at different points in its trajectory.<\/li>\n<li>Apply the principle of independence of motion to solve projectile motion problems.<\/li>\n<\/ul>\n<\/div>\n<p id=\"import-auto-id1846742\"><span data-type=\"term\" id=\"import-auto-id1560216\">Projectile motion<\/span> is the <span data-type=\"term\" id=\"import-auto-id1846113\">motion<\/span> of an object thrown or projected into the air, subject to only the acceleration of gravity. The object is called a <span data-type=\"term\" id=\"import-auto-id1809247\">projectile<\/span>, and its path is called its <span data-type=\"term\" id=\"import-auto-id1397020\">trajectory<\/span>. The motion of falling objects, as covered in <a href=\"\/contents\/68db177b-ab5c-48b9-8bcc-721e140fed8b@2\">Problem-Solving Basics for One-Dimensional Kinematics<\/a>, is a simple one-dimensional type of projectile motion in which there is no horizontal movement. In this section, we consider two-dimensional projectile motion, such as that of a football or other object for which <span data-type=\"term\" id=\"import-auto-id1230666\">air resistance<\/span> <em data-effect=\"italics\"><em data-effect=\"italics\">is negligible<\/em><\/em>.<\/p>\n<p id=\"import-auto-id1696126\">The most important fact to remember here is that <em data-effect=\"italics\"><em data-effect=\"italics\">motions along perpendicular axes are independent<\/em><\/em> and thus can be analyzed separately. This fact was discussed in <a href=\"\/contents\/21d0e217-d50f-4901-af75-905e738eb4c4@4\">Kinematics in Two Dimensions: An Introduction<\/a>, where vertical and horizontal motions were seen to be independent. The key to analyzing two-dimensional projectile motion is to break it into two motions, one along the horizontal axis and the other along the vertical. (This choice of axes is the most sensible, because acceleration due to gravity is vertical\u2014thus, there will be no acceleration along the horizontal axis when air resistance is negligible.) As is customary, we call the horizontal axis the <em data-effect=\"italics\"><em data-effect=\"italics\">x<\/em><\/em>-axis and the vertical axis the <em data-effect=\"italics\"><em data-effect=\"italics\">y<\/em><\/em>-axis. <a href=\"#import-auto-id2242290\" class=\"autogenerated-content\">(Figure)<\/a> illustrates the notation for displacement, where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-233158014558f77be2eb90341d930fba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\" \/> is defined to be the total displacement and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-bcda923e732ff6e429d93d0fa7ea8a47_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4ef20b6cb982fb0a79ab8a23ea132d96_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"11\" style=\"vertical-align: -3px;\" \/> are its components along the horizontal and vertical axes, respectively. The magnitudes of these vectors are <em data-effect=\"italics\"><em data-effect=\"italics\">s<\/em><\/em>, <em data-effect=\"italics\"><em data-effect=\"italics\">x<\/em><\/em>, and <em data-effect=\"italics\"><em data-effect=\"italics\">y<\/em><\/em>. (Note that in the last section we used the notation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-82a0de169ca718071f87c1bb7d571c4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"15\" style=\"vertical-align: -1px;\" \/> to represent a vector with components <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c204c4e308145f203183bde695b4b9af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#65;&#125;&#125;&#95;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"23\" style=\"vertical-align: -3px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-712cbf606ff38be790b1503be0cb9c9c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#65;&#125;&#125;&#95;&#123;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"22\" style=\"vertical-align: -6px;\" \/>. If we continued this format, we would call displacement <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-233158014558f77be2eb90341d930fba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\" \/> with components <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-34b9bb1fddbbb571283d7194014f0712_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#115;&#125;&#125;&#95;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"16\" style=\"vertical-align: -3px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0afa8f6c07a5872ef6a238deca66f787_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#115;&#125;&#125;&#95;&#123;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"15\" style=\"vertical-align: -6px;\" \/>. However, to simplify the notation, we will simply represent the component vectors as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-bcda923e732ff6e429d93d0fa7ea8a47_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4ef20b6cb982fb0a79ab8a23ea132d96_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"11\" style=\"vertical-align: -3px;\" \/>.)<\/p>\n<p id=\"import-auto-id1576953\">Of course, to describe motion we must deal with velocity and acceleration, as well as with displacement. We must find their components along the <em data-effect=\"italics\"><em data-effect=\"italics\">x<\/em><\/em>&#8211; and <em data-effect=\"italics\"><em data-effect=\"italics\">y<\/em><\/em>-axes, too. We will assume all forces except gravity (such as air resistance and friction, for example) are negligible. The components of acceleration are then very simple: <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d52b4af6d934e1dd5633b241c9dd7c5e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#121;&#125;&#61;&#45;&#103;&#61;&#45;&#57;&#46;&#56;&#48;&#32;&#109;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"171\" style=\"vertical-align: -6px;\" \/>. (Note that this definition assumes that the upwards direction is defined as the positive direction. If you arrange the coordinate system instead such that the downwards direction is positive, then acceleration due to gravity takes a positive value.) Because gravity is vertical, <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-547ef73deeca4f4dea21eb8c2fd93371_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#120;&#125;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"51\" style=\"vertical-align: -3px;\" \/>. Both accelerations are constant, so the kinematic equations can be used.<\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id1767845\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Review of Kinematic Equations (constant <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/>)<\/div>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-891\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-546f7811f3e4c495410b39135bddd277_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#123;&#120;&#125;&#95;&#123;&#48;&#125;&#43;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#45;&#125;&#123;&#118;&#125;&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"92\" style=\"vertical-align: -3px;\" \/><\/div>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-557\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5e6eca4279d03b3c094e9c9a366fb87a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#45;&#125;&#123;&#118;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#118;&#125;&#95;&#123;&#48;&#125;&#43;&#118;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"62\" style=\"vertical-align: -6px;\" \/><\/div>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9fc21b86018864d91df38dcdcb1d62eb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;&#61;&#123;&#118;&#125;&#95;&#123;&#48;&#125;&#43;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#116;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"86\" style=\"vertical-align: -3px;\" \/><\/div>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0093482a8f2ef19f27a21245075fde24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#123;&#120;&#125;&#95;&#123;&#48;&#125;&#43;&#123;&#118;&#125;&#95;&#123;&#48;&#125;&#116;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#116;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"151\" style=\"vertical-align: -6px;\" \/><\/div>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-198ba38595213db72748c6a0e6fcc2d9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#94;&#123;&#50;&#125;&#61;&#123;&#118;&#125;&#95;&#123;&#48;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#97;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#123;&#120;&#125;&#95;&#123;&#48;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"171\" style=\"vertical-align: -5px;\" \/><\/div>\n<\/div>\n<div class=\"bc-figure figure\" id=\"import-auto-id2242290\">\n<div class=\"bc-figcaption figcaption\">The total displacement <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-233158014558f77be2eb90341d930fba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\" \/> of a soccer ball at a point along its path. The vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-233158014558f77be2eb90341d930fba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\" \/> has components <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-bcda923e732ff6e429d93d0fa7ea8a47_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4ef20b6cb982fb0a79ab8a23ea132d96_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"11\" style=\"vertical-align: -3px;\" \/> along the horizontal and vertical axes. Its magnitude is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ae1901659f469e6be883797bfd30f4f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#115;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\" \/>, and it makes an angle <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-761998727948942ceb1b5763e45f01e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/> with the horizontal.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1402609\" data-alt=\"A soccer player is kicking a soccer ball. The ball travels in a projectile motion and reaches a point whose vertical distance is y and horizontal distance is x. The displacement between the kicking point and the final point is s. The angle made by this displacement vector with x axis is theta.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_04_01.jpg\" data-media-type=\"image\/jpg\" alt=\"A soccer player is kicking a soccer ball. The ball travels in a projectile motion and reaches a point whose vertical distance is y and horizontal distance is x. The displacement between the kicking point and the final point is s. The angle made by this displacement vector with x axis is theta.\" width=\"350\" \/><\/span><\/p>\n<\/div>\n<p id=\"eip-36\">Given these assumptions, the following steps are then used to analyze projectile motion:<\/p>\n<p id=\"eip-822\"><em data-effect=\"italics\"><strong>Step 1.<\/strong><\/em><em data-effect=\"italics\">Resolve or break the motion into horizontal and vertical components along the x- and y-axes.<\/em> These axes are perpendicular, so <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-63ab5ac3fe943d7cfd86c538c9aeb2c5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#65;&#125;&#95;&#123;&#120;&#125;&#61;&#65;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#111;&#115;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"100\" style=\"vertical-align: -3px;\" \/> and <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a81700950d2290414044809ca64a3d35_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#65;&#125;&#95;&#123;&#121;&#125;&#61;&#65;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#105;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"98\" style=\"vertical-align: -6px;\" \/> are used. The magnitude of the components of displacement <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-233158014558f77be2eb90341d930fba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\" \/> along these axes are <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/> and <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1f46f809fc7cf75ea75a335870507593_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#121;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: -3px;\" \/> The magnitudes of the components of the velocity <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1fb42cb2b0083b64704052f6366c336f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#118;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"9\" width=\"11\" style=\"vertical-align: 0px;\" \/> are <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-934e626b4207eec3b567077dd9b318de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#120;&#125;&#61;&#118;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#111;&#115;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"92\" style=\"vertical-align: -3px;\" \/> and <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-cfaad5937f6b5777e0e030f501bb42ff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#121;&#125;&#61;&#118;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#105;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#44;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"93\" style=\"vertical-align: -6px;\" \/> where <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ef71511c70f0e4b25cc6bd69f3bc20c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/> is the magnitude of the velocity and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-761998727948942ceb1b5763e45f01e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/> is its direction, as shown in <a href=\"#import-auto-id1815222\" class=\"autogenerated-content\">(Figure)<\/a>. Initial values are denoted with a subscript 0, as usual.<\/p>\n<p id=\"eip-205\"><em data-effect=\"italics\"><strong>Step 2.<\/strong><\/em><em data-effect=\"italics\">Treat the motion as two independent one-dimensional motions, one horizontal and the other vertical.<\/em> The kinematic equations for horizontal and vertical motion take the following forms:\n<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-338\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1d35d7f5f58f4b87602d574f8544a3e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#72;&#111;&#114;&#105;&#122;&#111;&#110;&#116;&#97;&#108;&#32;&#77;&#111;&#116;&#105;&#111;&#110;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#97;&#125;&#95;&#123;&#120;&#125;&#61;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"209\" style=\"vertical-align: -4px;\" \/><\/div>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-362\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-e4319d38b6e7f356737fa11a5400a6ed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#123;&#120;&#125;&#95;&#123;&#48;&#125;&#43;&#123;&#118;&#125;&#95;&#123;&#120;&#125;&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"96\" style=\"vertical-align: -3px;\" \/><\/div>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-627\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-04deafddb623681da0043ac93c319018_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#120;&#125;&#61;&#123;&#118;&#125;&#95;&#123;&#48;&#120;&#125;&#61;&#123;&#118;&#125;&#95;&#123;&#120;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#118;&#101;&#108;&#111;&#99;&#105;&#116;&#121;&#32;&#105;&#115;&#32;&#97;&#32;&#99;&#111;&#110;&#115;&#116;&#97;&#110;&#116;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"297\" style=\"vertical-align: -3px;\" \/><\/div>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-293\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f3a7a6569fda575839448de3d1f172d5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#86;&#101;&#114;&#116;&#105;&#99;&#97;&#108;&#32;&#77;&#111;&#116;&#105;&#111;&#110;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#115;&#115;&#117;&#109;&#105;&#110;&#103;&#32;&#112;&#111;&#115;&#105;&#116;&#105;&#118;&#101;&#32;&#105;&#115;&#32;&#117;&#112;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#97;&#125;&#95;&#123;&#121;&#125;&#61;&#45;&#103;&#61;&#45;&#57;&#46;&#92;&#116;&#101;&#120;&#116;&#123;&#56;&#48;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"499\" style=\"vertical-align: -12px;\" \/><\/div>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-131\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-58f67fa2a77742174edb305d7fc72bbc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#123;&#121;&#125;&#95;&#123;&#48;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#118;&#125;&#95;&#123;&#48;&#121;&#125;&#43;&#123;&#118;&#125;&#95;&#123;&#121;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"169\" style=\"vertical-align: -6px;\" \/><\/div>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-305\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-11415c888af5904daae5fa1c33404a5f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#121;&#125;&#61;&#123;&#118;&#125;&#95;&#123;&#48;&#121;&#125;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#103;&#116;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"101\" style=\"vertical-align: -6px;\" \/><\/div>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-542\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5e87a810bd1a6674c9a0bcf23118de82_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#123;&#121;&#125;&#95;&#123;&#48;&#125;&#43;&#123;&#118;&#125;&#95;&#123;&#48;&#121;&#125;&#116;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#103;&#116;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"156\" style=\"vertical-align: -6px;\" \/><\/div>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-243\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a11e6b73d37f00b90afe526f6c171463_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#123;&#118;&#125;&#95;&#123;&#48;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#123;&#121;&#125;&#95;&#123;&#48;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"175\" style=\"vertical-align: -8px;\" \/><\/div>\n<p id=\"eip-708\"><em data-effect=\"italics\"><strong>Step 3.<\/strong><\/em><em data-effect=\"italics\"> Solve for the unknowns in the two separate motions\u2014one horizontal and one vertical.<\/em> Note that the only common variable between the motions is time <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b4e3cbf5d4c5c6d9b702dd139f14c147_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"6\" style=\"vertical-align: 0px;\" \/>. The problem solving procedures here are the same as for one-dimensional <span data-type=\"term\">kinematics<\/span> and are illustrated in the solved examples below.<\/p>\n<p id=\"eip-979\"><em data-effect=\"italics\"><strong>Step 4.<\/strong><\/em><em data-effect=\"italics\">Recombine the two motions to find the total displacement<\/em><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-523ae81901d5840d93bce3c4df6f4d92_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"7\" style=\"vertical-align: 0px;\" \/><em data-effect=\"italics\"> and velocity <\/em><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b2a0ddecb4ad20c4420849c699255b2d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#118;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"9\" width=\"9\" style=\"vertical-align: 0px;\" \/>. Because the <em data-effect=\"italics\">x<\/em> &#8211; and <em data-effect=\"italics\">y<\/em> -motions are perpendicular, we determine these vectors by using the techniques outlined in the <a href=\"\/contents\/b9739bfd-dc9d-4f0a-b037-dc22884d30f3@10\">Vector Addition and Subtraction: Analytical Methods<\/a> and employing <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3ec0f49b86554b63597314fbe27102ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#123;&#65;&#125;&#95;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#65;&#125;&#95;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"120\" style=\"vertical-align: -13px;\" \/><br \/>\n        and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7d32b5632bd103155330185f116cfc82_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#97;&#110;&#125;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#65;&#125;&#95;&#123;&#121;&#125;&#47;&#123;&#65;&#125;&#95;&#123;&#120;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"143\" style=\"vertical-align: -6px;\" \/> in the following form, where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-761998727948942ceb1b5763e45f01e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/> is the direction of the displacement <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-233158014558f77be2eb90341d930fba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-26aa6e7bf6a33468d79e9df43deb3d1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#95;&#123;&#118;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"15\" style=\"vertical-align: -3px;\" \/> is the direction of the velocity <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1fb42cb2b0083b64704052f6366c336f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#118;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"9\" width=\"11\" style=\"vertical-align: 0px;\" \/>:\n<\/p>\n<p id=\"eip-245\"><strong>Total displacement and velocity<\/strong><\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-743\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7201fd7456359b53ddb4db2b3bf6e95f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#115;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"107\" style=\"vertical-align: -5px;\" \/><\/div>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-373\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-fdc77ab27e1af3adc6bceb7616e9f5e6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#97;&#110;&#125;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#47;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"121\" style=\"vertical-align: -5px;\" \/><\/div>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-679\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-bfe3b8dc3ed5fc70f5bc0ac42f1e4050_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#123;&#118;&#125;&#95;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#118;&#125;&#95;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"107\" style=\"vertical-align: -13px;\" \/><\/div>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-264\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7652e7e48d199c50f549c695240c2431_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#95;&#123;&#118;&#125;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#97;&#110;&#125;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#118;&#125;&#95;&#123;&#121;&#125;&#47;&#123;&#118;&#125;&#95;&#123;&#120;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"150\" style=\"vertical-align: -6px;\" \/><\/div>\n<div class=\"bc-figure figure\" id=\"import-auto-id1815222\">\n<div class=\"bc-figcaption figcaption\">(a) We analyze two-dimensional projectile motion by breaking it into two independent one-dimensional motions along the vertical and horizontal axes. (b) The horizontal motion is simple, because <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-547ef73deeca4f4dea21eb8c2fd93371_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#120;&#125;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"51\" style=\"vertical-align: -3px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c729d9ba572764ed37d6443270d2e745_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"17\" style=\"vertical-align: -3px;\" \/> is thus constant. (c) The velocity in the vertical direction begins to decrease as the object rises; at its highest point, the vertical velocity is zero. As the object falls towards the Earth again, the vertical velocity increases again in magnitude but points in the opposite direction to the initial vertical velocity. (d) The <em data-effect=\"italics\">x<\/em> &#8211; and <em data-effect=\"italics\">y<\/em> -motions are recombined to give the total velocity at any given point on the trajectory.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id2275311\" data-alt=\"In part a the figure shows projectile motion of a ball with initial velocity of v zero at an angle of theta zero with the horizontal x axis. The horizontal component v x and the vertical component v y at various positions of ball in the projectile path is shown. In part b only the horizontal velocity component v sub x is shown whose magnitude is constant at various positions in the path. In part c only vertical velocity component v sub y is shown. The vertical velocity component v sub y is upwards till it reaches the maximum point and then its direction changes to downwards. In part d resultant v of horizontal velocity component v sub x and downward vertical velocity component v sub y is found which makes an angle theta with the horizontal x axis. The direction of resultant velocity v is towards south east.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_04_02.jpg\" data-media-type=\"image\/jpg\" alt=\"In part a the figure shows projectile motion of a ball with initial velocity of v zero at an angle of theta zero with the horizontal x axis. The horizontal component v x and the vertical component v y at various positions of ball in the projectile path is shown. In part b only the horizontal velocity component v sub x is shown whose magnitude is constant at various positions in the path. In part c only vertical velocity component v sub y is shown. The vertical velocity component v sub y is upwards till it reaches the maximum point and then its direction changes to downwards. In part d resultant v of horizontal velocity component v sub x and downward vertical velocity component v sub y is found which makes an angle theta with the horizontal x axis. The direction of resultant velocity v is towards south east.\" height=\"600\" \/><\/span><\/p>\n<\/div>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id2175010\">\n<div data-type=\"title\" class=\"title\">A Fireworks Projectile Explodes High and Away<\/div>\n<p id=\"import-auto-id1896064\">During a fireworks display, a shell is shot into the air with an initial speed of 70.0 m\/s at an angle of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-faadb5ae3dfae0fb6515c4f193d500e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#53;&#46;&#48;&ordm;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"32\" style=\"vertical-align: 0px;\" \/> above the horizontal, as illustrated in <a href=\"#import-auto-id934168\" class=\"autogenerated-content\">(Figure)<\/a>. The fuse is timed to ignite the shell just as it reaches its highest point above the ground. (a) Calculate the height at which the shell explodes. (b) How much time passed between the launch of the shell and the explosion? (c) What is the horizontal displacement of the shell when it explodes?<\/p>\n<p id=\"eip-149\"><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id1629969\">Because air resistance is negligible for the unexploded shell, the analysis method outlined above can be used. The motion can be broken into horizontal and vertical motions in which  <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-547ef73deeca4f4dea21eb8c2fd93371_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#120;&#125;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"51\" style=\"vertical-align: -3px;\" \/> and  <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-bfbd41e51161676df80b591bcd64d1cb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#121;&#125;&#61;&#45;&#103;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"64\" style=\"vertical-align: -6px;\" \/>. We can then define <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-96b224e50cd20bf6c24005d45c5a085c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"17\" style=\"vertical-align: -3px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5f7ed76541634323a26ed2f60c7ac1c9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: -4px;\" \/> to be zero and solve for the desired quantities.<\/p>\n<p id=\"eip-774\"><strong>Solution for (a)<\/strong><\/p>\n<p id=\"import-auto-id1669571\">By \u201cheight\u201d we mean the altitude or vertical position <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/> above the starting point. The highest point in any trajectory, called the apex, is reached when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4da8b1b08d5d79b68e1c48d4683fe09b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#121;&#125;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"49\" style=\"vertical-align: -6px;\" \/>. Since we know the initial and final velocities as well as the initial position, we use the following equation to find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/>:<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-734\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a11e6b73d37f00b90afe526f6c171463_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#123;&#118;&#125;&#95;&#123;&#48;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#123;&#121;&#125;&#95;&#123;&#48;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"175\" style=\"vertical-align: -8px;\" \/><\/div>\n<div class=\"bc-figure figure\" id=\"import-auto-id934168\">\n<div class=\"bc-figcaption figcaption\">The trajectory of a fireworks shell. The fuse is set to explode the shell at the highest point in its trajectory, which is found to be at a height of 233 m and 125 m away horizontally.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1576299\" data-alt=\"The x y graph shows the trajectory of fireworks shell. The initial velocity of the shell v zero is at angle theta zero equal to seventy five degrees with the horizontal x axis. The fuse is set to explode the shell at the highest point of the trajectory which is at a height h equal to two hundred thirty three meters and at a horizontal distance x equal to one hundred twenty five meters from the origin.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_04_03a.jpg\" data-media-type=\"image\/jpg\" alt=\"The x y graph shows the trajectory of fireworks shell. The initial velocity of the shell v zero is at angle theta zero equal to seventy five degrees with the horizontal x axis. The fuse is set to explode the shell at the highest point of the trajectory which is at a height h equal to two hundred thirty three meters and at a horizontal distance x equal to one hundred twenty five meters from the origin.\" height=\"250\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id1163607\">Because <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5f7ed76541634323a26ed2f60c7ac1c9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-48e06402d9a14caec090bcc980729dfc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"16\" style=\"vertical-align: -6px;\" \/> are both zero, the equation simplifies to<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a9d5f8ea204e689604bad86f506df866_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#61;&#123;&#118;&#125;&#95;&#123;&#48;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#92;&#116;&#101;&#120;&#116;&#123;&#103;&#121;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"107\" style=\"vertical-align: -8px;\" \/><\/div>\n<p id=\"import-auto-id2114969\">Solving for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/> gives<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-256\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9c2c5c14ec9b6088a823af9e8eb1f4bb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#118;&#125;&#95;&#123;&#48;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#50;&#103;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"31\" width=\"61\" style=\"vertical-align: -9px;\" \/><\/div>\n<p id=\"import-auto-id1877237\">Now we must find <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-e36e9d4d1aad7476aff65cc15271ed2e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#48;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"23\" style=\"vertical-align: -6px;\" \/>, the component of the initial velocity in the <em data-effect=\"italics\">y<\/em>-direction. It is given by <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-08e4fce9903a4d9d7d404dacb71ce935_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#48;&#121;&#125;&#61;&#123;&#118;&#125;&#95;&#123;&#123;&#48;&#125;&#94;&#123;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#105;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"104\" style=\"vertical-align: -6px;\" \/>, where <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-e36e9d4d1aad7476aff65cc15271ed2e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#48;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"23\" style=\"vertical-align: -6px;\" \/> is the initial velocity of 70.0 m\/s, and <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-edb37b22a5621eda0f412ba51bb0745d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#95;&#123;&#48;&#125;&#61;&#55;&#53;&#46;&#48;&ordm;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"72\" style=\"vertical-align: -3px;\" \/> is the initial angle. Thus,<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-677\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-53c92c3992f00cbba32d33df01cdc14d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#48;&#121;&#125;&#61;&#123;&#118;&#125;&#95;&#123;&#48;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#105;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#95;&#123;&#48;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#55;&#48;&#46;&#48;&#32;&#109;&#47;&#115;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#105;&#110;&#32;&#55;&#53;&ordm;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#55;&#46;&#54;&#32;&#109;&#47;&#115;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"374\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id2271493\">and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/> is<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-512\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-974cdd8b63edd026ef12d65186d17f26_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#55;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#54;&#32;&#109;&#47;&#115;&#125;&#123;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#56;&#48;&#32;&#109;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"34\" width=\"121\" style=\"vertical-align: -14px;\" \/><\/div>\n<p id=\"import-auto-id1919502\">so that<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-310\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-38638307228d2b7d30dfa43ca527dfcb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#51;&#51;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"79\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"eip-429\"><strong>Discussion for (a)<\/strong><\/p>\n<p id=\"import-auto-id2165781\">Note that because up is positive, the initial velocity is positive, as is the maximum height, but the acceleration due to gravity is negative. Note also that the maximum height depends only on the vertical component of the initial velocity, so that any projectile with a 67.6 m\/s initial vertical component of velocity will reach a maximum height of 233 m (neglecting air resistance). The numbers in this example are reasonable for large fireworks displays, the shells of which do reach such heights before exploding. In practice, air resistance is not completely negligible, and so the initial velocity would have to be somewhat larger than that given to reach the same height.<\/p>\n<p id=\"eip-449\"><strong>Solution for (b)<\/strong><\/p>\n<p id=\"import-auto-id1632028\">As in many physics problems, there is more than one way to solve for the time to the highest point. In this case, the easiest method is to use <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-58f67fa2a77742174edb305d7fc72bbc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#123;&#121;&#125;&#95;&#123;&#48;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#118;&#125;&#95;&#123;&#48;&#121;&#125;&#43;&#123;&#118;&#125;&#95;&#123;&#121;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"169\" style=\"vertical-align: -6px;\" \/>. Because <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5f7ed76541634323a26ed2f60c7ac1c9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: -4px;\" \/> is zero, this equation reduces to simply<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-383\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-bdedad63d559374e83bcbd42bca760a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#118;&#125;&#95;&#123;&#48;&#121;&#125;&#43;&#123;&#118;&#125;&#95;&#123;&#121;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#116;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"135\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id1383088\">Note that the final vertical velocity, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-48e06402d9a14caec090bcc980729dfc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"16\" style=\"vertical-align: -6px;\" \/>, at the highest point is zero. Thus,<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-50\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-665c255cea918f2b81cb810db6df1845_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#108;&#108;&#108;&#125;&#116;&#38;&#32;&#61;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#121;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#118;&#125;&#95;&#123;&#48;&#121;&#125;&#43;&#123;&#118;&#125;&#95;&#123;&#121;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#51;&#51;&#32;&#109;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#55;&#46;&#54;&#32;&#109;&#47;&#115;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#46;&#57;&#48;&#32;&#115;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"201\" style=\"vertical-align: -14px;\" \/><\/div>\n<p id=\"eip-31\"><strong>Discussion for (b)<\/strong><\/p>\n<p id=\"import-auto-id1848626\">This time is also reasonable for large fireworks. When you are able to see the launch of fireworks, you will notice several seconds pass before the shell explodes. (Another way of finding the time is by using <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-104497195a687c48c611310294e90df2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#123;&#121;&#125;&#95;&#123;&#48;&#125;&#43;&#123;&#118;&#125;&#95;&#123;&#48;&#121;&#125;&#116;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#103;&#116;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"156\" style=\"vertical-align: -6px;\" \/>, and solving the quadratic equation for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b4e3cbf5d4c5c6d9b702dd139f14c147_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"6\" style=\"vertical-align: 0px;\" \/>.)<\/p>\n<p id=\"eip-939\"><strong>Solution for (c)<\/strong><\/p>\n<p id=\"import-auto-id2262600\">Because air resistance is negligible, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-547ef73deeca4f4dea21eb8c2fd93371_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#120;&#125;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"51\" style=\"vertical-align: -3px;\" \/> and the horizontal velocity is constant, as discussed above. The horizontal displacement is horizontal velocity multiplied by time as given by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-e4319d38b6e7f356737fa11a5400a6ed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#123;&#120;&#125;&#95;&#123;&#48;&#125;&#43;&#123;&#118;&#125;&#95;&#123;&#120;&#125;&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"96\" style=\"vertical-align: -3px;\" \/>, where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-96b224e50cd20bf6c24005d45c5a085c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"17\" style=\"vertical-align: -3px;\" \/> is equal to zero:<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-675\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f9933c41fc6e38be812cad9c3cd76f3f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#123;&#118;&#125;&#95;&#123;&#120;&#125;&#116;&#92;&#116;&#101;&#120;&#116;&#123;&#44;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"61\" style=\"vertical-align: -3px;\" \/><\/div>\n<p id=\"import-auto-id1871833\">where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c729d9ba572764ed37d6443270d2e745_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"17\" style=\"vertical-align: -3px;\" \/> is the <em data-effect=\"italics\"><em data-effect=\"italics\">x<\/em><\/em>-component of the velocity, which is given by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6cd820a1602bd24a01eef2c6946f050b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#120;&#125;&#61;&#123;&#118;&#125;&#95;&#123;&#48;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#111;&#115;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#95;&#123;&#48;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"110\" style=\"vertical-align: -3px;\" \/> Now,<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-884\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-58c9482c9610d39e871ed50ad413b05b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#120;&#125;&#61;&#123;&#118;&#125;&#95;&#123;&#48;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#111;&#115;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#95;&#123;&#48;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#55;&#48;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&#32;&#109;&#47;&#115;&#92;&#116;&#101;&#120;&#116;&#123;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#111;&#115;&#32;&#55;&#53;&#46;&#48;&ordm;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#56;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#49;&#32;&#109;&#47;&#115;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"378\" style=\"vertical-align: -5px;\" \/><\/div>\n<p id=\"import-auto-id2046887\">The time <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b4e3cbf5d4c5c6d9b702dd139f14c147_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"6\" style=\"vertical-align: 0px;\" \/> for both motions is the same, and so <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/> is<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-685\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f640a19bfc97379bbf588aac869c2901_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#56;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#49;&#32;&#109;&#47;&#115;&#92;&#116;&#101;&#120;&#116;&#123;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#48;&#32;&#115;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#50;&#53;&#32;&#109;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"248\" style=\"vertical-align: -5px;\" \/><\/div>\n<p id=\"eip-247\"><strong>Discussion for (c)<\/strong><\/p>\n<p id=\"import-auto-id1857186\">The horizontal motion is a constant velocity in the absence of air resistance. The horizontal displacement found here could be useful in keeping the fireworks fragments from falling on spectators. Once the shell explodes, air resistance has a major effect, and many fragments will land directly below.<\/p>\n<\/div>\n<p id=\"import-auto-id1986266\">In solving part (a) of the preceding example, the expression we found for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/> is valid for any projectile motion where air resistance is negligible. Call the maximum height <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-dfeace4a19a0ca888b9eac06eda60987_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#104;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"43\" style=\"vertical-align: -4px;\" \/>; then,<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-803\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6fbb0fc5cd33056d5ce848ee29eec587_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#118;&#125;&#95;&#123;&#48;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#50;&#103;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"31\" width=\"62\" style=\"vertical-align: -9px;\" \/><\/div>\n<p id=\"import-auto-id1973673\">This equation defines the <em data-effect=\"italics\"><em data-effect=\"italics\">maximum height of a projectile<\/em><\/em> and depends only on the vertical component of the initial velocity.<\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id1479427\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Defining a Coordinate System<\/div>\n<p id=\"import-auto-id2275341\">It is important to set up a coordinate system when analyzing projectile motion. One part of defining the coordinate system is to define an origin for the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/> positions. Often, it is convenient to choose the initial position of the object as the origin such that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-42f018c4cd0cdc1ab9c64865952e35f2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#95;&#123;&#48;&#125;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"50\" style=\"vertical-align: -3px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9154904d3603ba85ab0697f791ea4b68_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#95;&#123;&#48;&#125;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"49\" style=\"vertical-align: -4px;\" \/>. It is also important to define the positive and negative directions in the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/> directions. Typically, we define the positive vertical direction as upwards, and the positive horizontal direction is usually the direction of the object\u2019s motion. When this is the case, the vertical acceleration, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d208fd391fa57c168dc0f151de829fee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/>, takes a negative value (since it is directed downwards towards the Earth). However, it is occasionally useful to define the coordinates differently. For example, if you are analyzing the motion of a ball thrown downwards from the top of a cliff, it may make sense to define the positive direction downwards since the motion of the ball is solely in the downwards direction. If this is the case, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d208fd391fa57c168dc0f151de829fee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/> takes a positive value.<\/p>\n<\/div>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id708626\">\n<div data-type=\"title\" class=\"title\">Calculating Projectile Motion: Hot Rock Projectile<\/div>\n<p id=\"import-auto-id1916950\">Kilauea in Hawaii is the world\u2019s most continuously active volcano. Very active volcanoes characteristically eject red-hot rocks and lava rather than smoke and ash. Suppose a large rock is ejected from the volcano with a speed of 25.0 m\/s and at an angle <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1ee07f569fc6e62eba51a6a462cc4446_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#53;&#46;&#48;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"32\" style=\"vertical-align: 0px;\" \/> above the horizontal, as shown in <a href=\"#import-auto-id1817519\" class=\"autogenerated-content\">(Figure)<\/a>. The rock strikes the side of the volcano at an altitude 20.0 m lower than its starting point. (a) Calculate the time it takes the rock to follow this path. (b) What are the magnitude and direction of the rock\u2019s velocity at impact?<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1817519\">\n<div class=\"bc-figcaption figcaption\">The trajectory of a rock ejected from the Kilauea volcano.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1817520\" data-alt=\"The trajectory of a rock ejected from a volcano is shown. The initial velocity of rock v zero is equal to twenty five meters per second and it makes an angle of thirty five degrees with the horizontal x axis. The figure shows rock falling down a height of twenty meters below the volcano level. The velocity at this point is v which makes an angle of theta with horizontal x axis. The direction of v is south east.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_04_04a.jpg\" data-media-type=\"image\/jpg\" alt=\"The trajectory of a rock ejected from a volcano is shown. The initial velocity of rock v zero is equal to twenty five meters per second and it makes an angle of thirty five degrees with the horizontal x axis. The figure shows rock falling down a height of twenty meters below the volcano level. The velocity at this point is v which makes an angle of theta with horizontal x axis. The direction of v is south east.\" width=\"400\" \/><\/span><\/p>\n<\/div>\n<p id=\"eip-770\"><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id1823692\">Again, resolving this two-dimensional motion into two independent one-dimensional motions will allow us to solve for the desired quantities. The time a projectile is in the air is governed by its vertical motion alone. We will solve for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b4e3cbf5d4c5c6d9b702dd139f14c147_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"6\" style=\"vertical-align: 0px;\" \/> first. While the rock is rising and falling vertically, the horizontal motion continues at a constant velocity. This example asks for the final velocity. Thus, the vertical and horizontal results will be recombined to obtain <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ef71511c70f0e4b25cc6bd69f3bc20c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-26aa6e7bf6a33468d79e9df43deb3d1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#95;&#123;&#118;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"15\" style=\"vertical-align: -3px;\" \/> at the final time <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b4e3cbf5d4c5c6d9b702dd139f14c147_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"6\" style=\"vertical-align: 0px;\" \/> determined in the first part of the example.<\/p>\n<p id=\"eip-408\"><strong>Solution for (a)<\/strong><\/p>\n<p id=\"import-auto-id1608071\">While the rock is in the air, it rises and then falls to a final position 20.0 m lower than its starting altitude. We can find the time for this by using<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-895\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1042c5f7bc6b35ebf926f1e8a5d05a8c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#123;&#121;&#125;&#95;&#123;&#48;&#125;&#43;&#123;&#118;&#125;&#95;&#123;&#48;&#121;&#125;&#116;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#103;&#116;&#125;&#125;&#94;&#123;&#50;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"161\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id1843834\">If we take the initial position <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5f7ed76541634323a26ed2f60c7ac1c9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: -4px;\" \/> to be zero, then the final position is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-fa7cb1a352e3aaecdf3962965dffdd86_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#48;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#48;&#32;&#109;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"103\" style=\"vertical-align: -4px;\" \/> Now the initial vertical velocity is the vertical component of the initial velocity, found from  <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5d9e3121f60dcf9cf3e3c2a2f8e5a938_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#48;&#121;&#125;&#61;&#123;&#118;&#125;&#95;&#123;&#48;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#105;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"109\" style=\"vertical-align: -6px;\" \/> = (<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-fdb7d42823227589ecf1d0a8833d7545_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#53;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#32;&#109;&#47;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"68\" style=\"vertical-align: -4px;\" \/>)(<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a8cfc5daca5b2033d9830bbaa8b01d9a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#105;&#110;&#32;&#51;&#53;&#46;&#48;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"60\" style=\"vertical-align: 0px;\" \/>) = <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5a7ad0f248b80551b4bb227ff6a97015_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#52;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#32;&#109;&#47;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"67\" style=\"vertical-align: -4px;\" \/>. Substituting known values yields<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-fb4b9e41d4edea97cf99b094043f60e3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#48;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&#32;&#109;&#92;&#116;&#101;&#120;&#116;&#123;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#52;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#51;&#32;&#109;&#47;&#115;&#92;&#116;&#101;&#120;&#116;&#123;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#116;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#48;&#32;&#109;&#47;&#115;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#125;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#123;&#116;&#125;&#94;&#123;&#50;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"306\" style=\"vertical-align: -7px;\" \/><\/div>\n<p id=\"import-auto-id1561988\">Rearranging terms gives a quadratic equation in <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b4e3cbf5d4c5c6d9b702dd139f14c147_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"6\" style=\"vertical-align: 0px;\" \/>:<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-931\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-03cab08ed95664aa9ff4f56cdf1d8171_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#48;&#32;&#109;&#47;&#115;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#125;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#123;&#116;&#125;&#94;&#123;&#50;&#125;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#52;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#32;&#109;&#47;&#115;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#116;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#48;&#46;&#48;&#32;&#109;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"344\" style=\"vertical-align: -7px;\" \/><\/div>\n<p id=\"import-auto-id2244917\">This expression is a quadratic equation of the form<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-abbc666a660e806c2ac4d57b195367cb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#97;&#116;&#125;&#125;&#94;&#123;&#50;&#125;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#98;&#116;&#125;&#43;&#99;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"124\" style=\"vertical-align: -2px;\" \/>, where the constants are <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-326b073ca925269813fcf24c033b0ffb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#52;&#46;&#57;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"65\" style=\"vertical-align: -1px;\" \/>, <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a5cdc9c6effa11e3c88cc8361d9bbc0c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#61;&#45;&#49;&#52;&#46;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"77\" style=\"vertical-align: -1px;\" \/>, and <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-8bb4419b7ad6f07e72340fbc322b45b9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#61;&#45;&#50;&#48;&#46;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"81\" style=\"vertical-align: 0px;\" \/> Its solutions are given by the quadratic formula:<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-880\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a5b1a7da3632fc3b1a504581feec0383_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#98;&plusmn;&#92;&#115;&#113;&#114;&#116;&#123;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#99;&#125;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"109\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id1600199\">This equation yields two solutions: <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0b2e28492b7b1b898cdbd822d935c7e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#61;&#51;&#46;&#57;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"62\" style=\"vertical-align: 0px;\" \/> and <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-71114c04a58e1ac4de3b44417238344c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#61;&#45;&#49;&#46;&#48;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"76\" style=\"vertical-align: -1px;\" \/>. (It is left as an exercise for the reader to verify these solutions.) The time is <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ff6b13ed1178ebbac38013fa459dd1e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#61;&#51;&#46;&#57;&#54;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"73\" style=\"vertical-align: 0px;\" \/> or <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d0e8c56c4613f8791ff31cb72e00f7b9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#46;&#48;&#51;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"56\" style=\"vertical-align: -1px;\" \/>. The negative value of time implies an event before the start of motion, and so we discard it. Thus,<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-267\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-77fa76897aacb9a935ffc0232112a9e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#61;&#51;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#54;&#32;&#115;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"78\" style=\"vertical-align: 0px;\" \/><\/div>\n<p id=\"eip-46\"><strong>Discussion for (a)<\/strong><\/p>\n<p id=\"import-auto-id1368589\">The time for projectile motion is completely determined by the vertical motion. So any projectile that has an initial vertical velocity of 14.3 m\/s and lands 20.0 m below its starting altitude will spend 3.96 s in the air.<\/p>\n<p id=\"eip-653\"><strong>Solution for (b)<\/strong><\/p>\n<p id=\"import-auto-id1599448\">From the information now in hand, we can find the final horizontal and vertical velocities <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c729d9ba572764ed37d6443270d2e745_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"17\" style=\"vertical-align: -3px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-48e06402d9a14caec090bcc980729dfc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"16\" style=\"vertical-align: -6px;\" \/> and combine them to find the total velocity <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ef71511c70f0e4b25cc6bd69f3bc20c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/> and the angle <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-31af75b2dc98204908b7f2afaa5695ea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"15\" style=\"vertical-align: -3px;\" \/> it makes with the horizontal. Of course, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c729d9ba572764ed37d6443270d2e745_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"17\" style=\"vertical-align: -3px;\" \/> is constant so we can solve for it at any horizontal location. In this case, we chose the starting point since we know both the initial velocity and initial angle. Therefore:<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-873\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-488a4433915fd8e290c126bedbb48890_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#120;&#125;&#61;&#123;&#118;&#125;&#95;&#123;&#48;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#111;&#115;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#95;&#123;&#48;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#53;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&#32;&#109;&#47;&#115;&#92;&#116;&#101;&#120;&#116;&#123;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#111;&#115;&#32;&#51;&#53;&ordm;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#48;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#53;&#32;&#109;&#47;&#115;&#46;&#92;&#116;&#101;&#120;&#116;&#123;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"364\" style=\"vertical-align: -5px;\" \/><\/div>\n<p id=\"import-auto-id2252925\">The final vertical velocity is given by the following equation:<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-168\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d289c5f73ad21c641123581c48e279f4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#121;&#125;&#61;&#123;&#118;&#125;&#95;&#123;&#48;&#121;&#125;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#103;&#116;&#44;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"106\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id2173689\">where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-e36e9d4d1aad7476aff65cc15271ed2e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#48;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"23\" style=\"vertical-align: -6px;\" \/> was found in part (a) to be <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5a7ad0f248b80551b4bb227ff6a97015_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#52;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#32;&#109;&#47;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"67\" style=\"vertical-align: -4px;\" \/>. Thus,<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-113\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-728b23b75b937d3c82ae39e6fff1bd13_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#121;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#52;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#51;&#32;&#109;&#47;&#115;&#92;&#116;&#101;&#120;&#116;&#123;&#125;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#56;&#48;&#32;&#109;&#47;&#115;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#125;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#54;&#32;&#115;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"278\" style=\"vertical-align: -7px;\" \/><\/div>\n<p id=\"import-auto-id1792451\">so that<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-571\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6e8b36658f87a201ac1c06d0e9ceaee4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#121;&#125;&#61;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#52;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#53;&#32;&#109;&#47;&#115;&#46;&#92;&#116;&#101;&#120;&#116;&#123;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"123\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id2239982\">To find the magnitude of the final velocity <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ef71511c70f0e4b25cc6bd69f3bc20c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/> we combine its perpendicular components, using the following equation:<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-394\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c8abd87365abcd1294b1583072525ef0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#123;&#118;&#125;&#95;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#118;&#125;&#95;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#48;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#53;&#32;&#109;&#47;&#115;&#92;&#116;&#101;&#120;&#116;&#123;&#125;&#123;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#52;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#53;&#32;&#109;&#47;&#115;&#92;&#116;&#101;&#120;&#116;&#123;&#125;&#123;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#44;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"362\" style=\"vertical-align: -13px;\" \/><\/div>\n<p id=\"import-auto-id1677955\">which gives<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-60\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2cba91ce009167f4db47b18a13d02c95_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#49;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#57;&#32;&#109;&#47;&#115;&#46;&#92;&#116;&#101;&#120;&#116;&#123;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"102\" style=\"vertical-align: -5px;\" \/><\/div>\n<p id=\"import-auto-id1645980\">The direction <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-26aa6e7bf6a33468d79e9df43deb3d1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#95;&#123;&#118;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"15\" style=\"vertical-align: -3px;\" \/> is found from the equation:<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-353\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7781402bc8622710c065bae19e67555a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#95;&#123;&#118;&#125;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#97;&#110;&#125;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#118;&#125;&#95;&#123;&#121;&#125;&#47;&#123;&#118;&#125;&#95;&#123;&#120;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"142\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id1972156\">so that<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-589\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a2f7a9e935028b190c462ab48cd0c7f0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#95;&#123;&#118;&#125;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#97;&#110;&#125;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#52;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#53;&#47;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#48;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#97;&#110;&#125;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#57;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"323\" style=\"vertical-align: -5px;\" \/><\/div>\n<p id=\"import-auto-id1613163\">Thus,<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1775885133d5b451af983e0cf38ff42e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#95;&#123;&#118;&#125;&#61;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#48;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#49;&ordm;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"89\" style=\"vertical-align: -3px;\" \/><\/div>\n<p id=\"eip-299\"><strong>Discussion for (b)<\/strong><\/p>\n<p id=\"import-auto-id1459619\">The negative angle means that the velocity is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5b12bc2fc2585ff0c1bcd89a1c45d462_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#48;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#49;&ordm;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"31\" style=\"vertical-align: -1px;\" \/> below the horizontal. This result is consistent with the fact that the final vertical velocity is negative and hence downward\u2014as you would expect because the final altitude is 20.0 m lower than the initial altitude. (See <a href=\"#import-auto-id1817519\" class=\"autogenerated-content\">(Figure)<\/a>.)<\/p>\n<\/div>\n<p id=\"import-auto-id1532818\">One of the most important things illustrated by projectile motion is that vertical and horizontal motions are independent of each other. Galileo was the first person to fully comprehend this characteristic. He used it to predict the range of a projectile. On level ground, we define <span data-type=\"term\" id=\"import-auto-id1751163\">range<\/span> to be the horizontal distance <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-dae6bae3dcdac4629730754352c5e329_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> traveled by a projectile. Galileo and many others were interested in the range of projectiles primarily for military purposes\u2014such as aiming cannons. However, investigating the range of projectiles can shed light on other interesting phenomena, such as the orbits of satellites around the Earth. Let us consider projectile range further.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1904800\">\n<div class=\"bc-figcaption figcaption\">Trajectories of projectiles on level ground. (a) The greater the initial speed <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c3b9ce7297f522a77c357066d17856a7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"16\" style=\"vertical-align: -3px;\" \/>, the greater the range for a given initial angle. (b) The effect of initial angle <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-31af75b2dc98204908b7f2afaa5695ea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"15\" style=\"vertical-align: -3px;\" \/> on the range of a projectile with a given initial speed. Note that the range is the same for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2d0913ff5fe480f07f7fecfc6f0108ac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#53;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"16\" style=\"vertical-align: -1px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-e1ebe0c87d44849909739b93b1341865_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#55;&#53;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"17\" style=\"vertical-align: 0px;\" \/>, although the maximum heights of those paths are different.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1904802\" data-alt=\"Part a of the figure shows three different trajectories of projectiles on level ground. In each case the projectiles makes an angle of forty five degrees with the horizontal axis. The first projectile of initial velocity thirty meters per second travels a horizontal distance of R equal to ninety one point eight meters. The second projectile of initial velocity forty meters per second travels a horizontal distance of R equal to one hundred sixty three meters. The third projectile of initial velocity fifty meters per second travels a horizontal distance of R equal to two hundred fifty five meters.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_04_05a.jpg\" data-media-type=\"image\/jpg\" alt=\"Part a of the figure shows three different trajectories of projectiles on level ground. In each case the projectiles makes an angle of forty five degrees with the horizontal axis. The first projectile of initial velocity thirty meters per second travels a horizontal distance of R equal to ninety one point eight meters. The second projectile of initial velocity forty meters per second travels a horizontal distance of R equal to one hundred sixty three meters. The third projectile of initial velocity fifty meters per second travels a horizontal distance of R equal to two hundred fifty five meters.\" height=\"300\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id1750749\">How does the initial velocity of a projectile affect its range? Obviously, the greater the initial speed <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c3b9ce7297f522a77c357066d17856a7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"16\" style=\"vertical-align: -3px;\" \/>, the greater the range, as shown in <a href=\"#import-auto-id1904800\" class=\"autogenerated-content\">(Figure)<\/a>(a). The initial angle <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-31af75b2dc98204908b7f2afaa5695ea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"15\" style=\"vertical-align: -3px;\" \/> also has a dramatic effect on the range, as illustrated in <a href=\"#import-auto-id1904800\" class=\"autogenerated-content\">(Figure)<\/a>(b). For a fixed initial speed, such as might be produced by a cannon, the maximum range is obtained with <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f159f48fe5af20e8e94b3b5de153fecf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#95;&#123;&#48;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#53;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"57\" style=\"vertical-align: -3px;\" \/>. This is true only for conditions neglecting air resistance. If air resistance is considered, the maximum angle is approximately <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c0fb64f1513d70c6e6b9d68d4fe0b531_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#56;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/>. Interestingly, for every initial angle except <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1142d1c44cfaf3459c45a3d6cc399899_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#53;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"17\" style=\"vertical-align: -1px;\" \/>, there are two angles that give the same range\u2014the sum of those angles is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0ef094c705f55f76b4993ff72af9e73f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#48;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/>. The range also depends on the value of the acceleration of gravity <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d208fd391fa57c168dc0f151de829fee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/>. The lunar astronaut Alan Shepherd was able to drive a golf ball a great distance on the Moon because gravity is weaker there. The range <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-dae6bae3dcdac4629730754352c5e329_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> of a projectile on <em data-effect=\"italics\"><em data-effect=\"italics\">level ground<\/em><\/em> for which air resistance is negligible is given by <\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-240\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2d311d5763e114cdc1f5c1702076b9c9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#118;&#125;&#95;&#123;&#48;&#125;&#94;&#123;&#50;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#105;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#50;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#95;&#123;&#48;&#125;&#125;&#123;&#103;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#44;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"105\" style=\"vertical-align: -9px;\" \/><\/div>\n<p id=\"import-auto-id1674836\">where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c3b9ce7297f522a77c357066d17856a7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"16\" style=\"vertical-align: -3px;\" \/> is the initial speed and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-31af75b2dc98204908b7f2afaa5695ea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"15\" style=\"vertical-align: -3px;\" \/> is the initial angle relative to the horizontal. The proof of this equation is left as an end-of-chapter problem (hints are given), but it does fit the major features of projectile range as described.<\/p>\n<p id=\"import-auto-id1686986\">When we speak of the range of a projectile on level ground, we assume that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-dae6bae3dcdac4629730754352c5e329_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> is very small compared with the circumference of the Earth. If, however, the range is large, the Earth curves away below the projectile and acceleration of gravity changes direction along the path. The range is larger than predicted by the range equation given above because the projectile has farther to fall than it would on level ground. (See <a href=\"#import-auto-id1645881\" class=\"autogenerated-content\">(Figure)<\/a>.) If the initial speed is great enough, the projectile goes into orbit.  This possibility was recognized centuries before it could be accomplished. When an object is in orbit, the Earth curves away from underneath the object at the same rate as it falls. The object thus falls continuously but never hits the surface. These and other aspects of orbital motion, such as the rotation of the Earth, will be covered analytically and in greater depth later in this text.<\/p>\n<p id=\"import-auto-id1645878\">Once again we see that thinking about one topic, such as the range of a projectile, can lead us to others, such as the Earth orbits. In  <a href=\"\/contents\/b64da007-2df2-4425-8f4b-e4d42ed423d9@11\">Addition of Velocities<\/a>, we will examine the addition of velocities, which is another important aspect of two-dimensional kinematics and will also yield insights beyond the immediate topic.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1645881\">\n<div class=\"bc-figcaption figcaption\">Projectile to satellite. In each case shown here, a projectile is launched from a very high tower to avoid air resistance. With increasing initial speed, the range increases and becomes longer than it would be on level ground because the Earth curves away underneath its path. With a large enough initial speed, orbit is achieved.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1825851\" data-alt=\"A figure of the Earth is shown and on top of it a very high tower is placed. A projectile satellite is launched from this very high tower with initial velocity of v zero in the horizontal direction. Several trajectories are shown with increasing range. A circular trajectory is shown indicating the satellite achieved its orbit and it is revolving around the Earth.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_04_06a.jpg\" data-media-type=\"image\/jpg\" alt=\"A figure of the Earth is shown and on top of it a very high tower is placed. A projectile satellite is launched from this very high tower with initial velocity of v zero in the horizontal direction. Several trajectories are shown with increasing range. A circular trajectory is shown indicating the satellite achieved its orbit and it is revolving around the Earth.\" width=\"200\" \/><\/span><\/p>\n<\/div>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"eip-89\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">PhET Explorations: Projectile Motion<\/div>\n<p id=\"eip-id1169738107387\">Blast a Buick out of a cannon! Learn about projectile motion by firing various objects. Set the angle, initial speed, and mass. Add air resistance. Make a game out of this simulation by trying to hit a target.<\/p>\n<div class=\"bc-figure figure\" id=\"eip-id1462984\">\n<div class=\"bc-figcaption figcaption\"><a href=\"\/resources\/5f60797f22fa74f2a6b84d7fe2c9f149f1109c19\/projectile-motion_en.jar\">Projectile Motion<\/a><\/div>\n<p><span data-type=\"media\" id=\"Phet_module_3.4\" data-alt=\"\"><a href=\"\/resources\/5f60797f22fa74f2a6b84d7fe2c9f149f1109c19\/projectile-motion_en.jar\" data-type=\"image\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/PhET_Icon.png\" data-media-type=\"image\/png\" alt=\"\" data-print=\"false\" width=\"450\" \/><\/a><span data-media-type=\"image\/png\" data-print=\"true\" data-src=\"\/resources\/075500ad9f71890a85fe3f7a4137ac08e2b7907c\/PhET_Icon.png\" data-type=\"image\"><\/span><\/span><\/p>\n<\/div>\n<\/div>\n<div class=\"section-summary\" data-depth=\"1\" id=\"fs-id1843457\">\n<h1 data-type=\"title\">Summary<\/h1>\n<ul id=\"fs-id2130996\">\n<li id=\"import-auto-id1677012\">Projectile motion is the motion of an object through the air that is subject only to the acceleration of gravity.<\/li>\n<li id=\"import-auto-id1786765\">To solve projectile motion problems, perform the following steps:\n<ol id=\"fs-id1842700\" data-number-style=\"arabic\" class=\"stepwise\">\n<li id=\"import-auto-id1830314\">Determine a coordinate system. Then, resolve the position and\/or velocity of the object in the horizontal and vertical components. The components of position <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-233158014558f77be2eb90341d930fba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\" \/> are given by the quantities <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/>, and the components of the velocity <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1fb42cb2b0083b64704052f6366c336f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#118;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"9\" width=\"11\" style=\"vertical-align: 0px;\" \/> are given by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-934e626b4207eec3b567077dd9b318de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#120;&#125;&#61;&#118;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#111;&#115;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"92\" style=\"vertical-align: -3px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3655c4f859ccbfb3ac8a0bb830875e18_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#121;&#125;&#61;&#118;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#105;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"89\" style=\"vertical-align: -6px;\" \/>, where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ef71511c70f0e4b25cc6bd69f3bc20c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/> is the magnitude of the velocity and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-761998727948942ceb1b5763e45f01e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/> is its direction.<\/li>\n<li id=\"import-auto-id1830316\">Analyze the motion of the projectile in the horizontal direction using the following equations:\n<div data-type=\"equation\" class=\"equation\" id=\"eip-898\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2311e987ab13df50ad9baee9957a1b73_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#72;&#111;&#114;&#105;&#122;&#111;&#110;&#116;&#97;&#108;&#32;&#109;&#111;&#116;&#105;&#111;&#110;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#97;&#125;&#95;&#123;&#120;&#125;&#61;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"207\" style=\"vertical-align: -4px;\" \/><\/div>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-236\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-e4319d38b6e7f356737fa11a5400a6ed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#123;&#120;&#125;&#95;&#123;&#48;&#125;&#43;&#123;&#118;&#125;&#95;&#123;&#120;&#125;&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"96\" style=\"vertical-align: -3px;\" \/><\/div>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-612\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2990b4d4e8bc11e166762ea90875cda6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#120;&#125;&#61;&#123;&#118;&#125;&#95;&#123;&#48;&#120;&#125;&#61;&#123;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#118;&#125;&#125;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#120;&#125;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#118;&#101;&#108;&#111;&#99;&#105;&#116;&#121;&#32;&#105;&#115;&#32;&#97;&#32;&#99;&#111;&#110;&#115;&#116;&#97;&#110;&#116;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"297\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/li>\n<li id=\"import-auto-id1939082\">Analyze the motion of the projectile in the vertical direction using the following equations:\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id1939084\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-38d790094436a66b6def1ef4d22fd7d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#86;&#101;&#114;&#116;&#105;&#99;&#97;&#108;&#32;&#109;&#111;&#116;&#105;&#111;&#110;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#115;&#115;&#117;&#109;&#105;&#110;&#103;&#32;&#112;&#111;&#115;&#105;&#116;&#105;&#118;&#101;&#32;&#100;&#105;&#114;&#101;&#99;&#116;&#105;&#111;&#110;&#32;&#105;&#115;&#32;&#117;&#112;&#59;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#97;&#125;&#95;&#123;&#121;&#125;&#61;&#45;&#103;&#61;&#45;&#57;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#56;&#48;&#32;&#109;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"587\" style=\"vertical-align: -12px;\" \/><\/div>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id1492830\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-58f67fa2a77742174edb305d7fc72bbc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#123;&#121;&#125;&#95;&#123;&#48;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#118;&#125;&#95;&#123;&#48;&#121;&#125;&#43;&#123;&#118;&#125;&#95;&#123;&#121;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"169\" style=\"vertical-align: -6px;\" \/><\/div>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2022844\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-11415c888af5904daae5fa1c33404a5f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#121;&#125;&#61;&#123;&#118;&#125;&#95;&#123;&#48;&#121;&#125;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#103;&#116;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"101\" style=\"vertical-align: -6px;\" \/><\/div>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id1677876\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-104497195a687c48c611310294e90df2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#123;&#121;&#125;&#95;&#123;&#48;&#125;&#43;&#123;&#118;&#125;&#95;&#123;&#48;&#121;&#125;&#116;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#103;&#116;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"156\" style=\"vertical-align: -6px;\" \/><\/div>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id1653540\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-dbd66f999785aa910503843941245688_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#123;&#118;&#125;&#95;&#123;&#48;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#123;&#121;&#125;&#95;&#123;&#48;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"175\" style=\"vertical-align: -8px;\" \/><\/div>\n<\/li>\n<li id=\"import-auto-id1552181\">Recombine the horizontal and vertical components of location and\/or velocity using the following equations:\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2092332\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7201fd7456359b53ddb4db2b3bf6e95f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#115;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"107\" style=\"vertical-align: -5px;\" \/><\/div>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2282348\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-fdc77ab27e1af3adc6bceb7616e9f5e6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#97;&#110;&#125;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#47;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"121\" style=\"vertical-align: -5px;\" \/><\/div>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2274748\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-bfe3b8dc3ed5fc70f5bc0ac42f1e4050_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#123;&#118;&#125;&#95;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#118;&#125;&#95;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"107\" style=\"vertical-align: -13px;\" \/><\/div>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id1979208\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a5b41c81da2ca1477a571dfdf75ee4c6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#118;&#125;&#125;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#97;&#110;&#125;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#118;&#125;&#95;&#123;&#121;&#125;&#47;&#123;&#118;&#125;&#95;&#123;&#120;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"150\" style=\"vertical-align: -6px;\" \/><\/div>\n<\/li>\n<\/ol>\n<\/li>\n<li id=\"import-auto-id1888635\">The maximum height <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-14b463d0ecd5b350ced6cf1d6a12eef3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"10\" style=\"vertical-align: 0px;\" \/> of a projectile launched with initial vertical velocity <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-e36e9d4d1aad7476aff65cc15271ed2e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#48;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"23\" style=\"vertical-align: -6px;\" \/> is given by\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id1534227\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0559351a9ce6edf676121dbf1a57f5cc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#118;&#125;&#95;&#123;&#48;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#50;&#103;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"31\" width=\"62\" style=\"vertical-align: -9px;\" \/><\/div>\n<\/li>\n<li id=\"import-auto-id1823593\">The maximum horizontal distance traveled by a projectile is called the <strong>range<\/strong>. The range <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-dae6bae3dcdac4629730754352c5e329_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> of a projectile on level ground launched at an angle <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-31af75b2dc98204908b7f2afaa5695ea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"15\" style=\"vertical-align: -3px;\" \/> above the horizontal with initial speed <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c3b9ce7297f522a77c357066d17856a7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"16\" style=\"vertical-align: -3px;\" \/> is given by\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id1951750\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b99c370e0c7079944974f0fae05e1ab7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#118;&#125;&#95;&#123;&#48;&#125;&#94;&#123;&#50;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#105;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#50;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#95;&#123;&#48;&#125;&#125;&#123;&#103;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"105\" style=\"vertical-align: -9px;\" \/><\/div>\n<\/li>\n<\/ul>\n<\/div>\n<div class=\"conceptual-questions\" data-depth=\"1\" id=\"fs-id2865659\" data-element-type=\"conceptual-questions\">\n<h1 data-type=\"title\">Conceptual Questions<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2183300\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2183302\">\n<p id=\"fs-id2298558\">Answer the following questions for projectile motion on level ground assuming negligible air resistance (the initial angle being neither <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a47cfc3a49ca80166a5e1f089c13ad7c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#48;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/> nor <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0ef094c705f55f76b4993ff72af9e73f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#48;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/>): (a) Is the velocity ever zero? (b) When is the velocity a minimum? A maximum? (c) Can the velocity ever be the same as the initial velocity at a time other than at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b7b41acc5cb99fb07aaa07b445eb2483_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"39\" style=\"vertical-align: 0px;\" \/>? (d) Can the speed ever be the same as the initial speed at a time other than at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b7b41acc5cb99fb07aaa07b445eb2483_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"39\" style=\"vertical-align: 0px;\" \/>?\n        <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1638420\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1638423\">\n<p id=\"fs-id1638424\">Answer the following questions for projectile motion on level ground assuming negligible air resistance (the initial angle being neither <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a47cfc3a49ca80166a5e1f089c13ad7c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#48;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/> nor <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0ef094c705f55f76b4993ff72af9e73f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#48;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/>): (a) Is the acceleration ever zero? (b) Is the acceleration ever in the same direction as a component of velocity? (c) Is the acceleration ever opposite in direction to a component of velocity?\n        <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2062475\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2062477\">\n<p id=\"fs-id2062478\">For a fixed initial speed, the range of a projectile is determined by the angle at which it is fired. For all but the maximum, there are two angles that give the same range. Considering factors that might affect the ability of an archer to hit a target, such as wind, explain why the smaller angle (closer to the horizontal) is preferable. When would it be necessary for the archer to use the larger angle? Why does the punter in a football game use the higher trajectory?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1875651\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1875652\">\n<p id=\"fs-id1875653\">During a lecture demonstration, a professor places two coins on the edge of a table. She then flicks one of the coins horizontally off the table, simultaneously nudging the other over the edge. Describe the subsequent motion of the two coins, in particular discussing whether they hit the floor at the same time.\n        <\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"problems-exercises\" data-depth=\"1\" id=\"fs-id1875655\" data-element-type=\"problems-exercises\">\n<h1 data-type=\"title\">Problems &amp; Exercises<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1923898\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1105650\">\n<p id=\"fs-id1105652\">A projectile is launched at ground level with an initial speed of 50.0 m\/s at an angle of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3e16ee995a39db0e4671513c4f1e853c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#48;&#46;&#48;&ordm;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"32\" style=\"vertical-align: 0px;\" \/> above the horizontal. It strikes a target above the ground 3.00 seconds later. What are the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/> distances from where the projectile was launched to where it lands?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1543587\">\n<p id=\"fs-id1915206\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ca577a96972b7c4def87bdbefa93dddf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#108;&#108;&#108;&#125;&#120;&#38;&#32;&#61;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#46;&#51;&#48;&#32;&#109;&#125;&times;&#123;&#49;&#48;&#125;&#94;&#123;&#50;&#125;&#92;&#92;&#32;&#121;&#38;&#32;&#61;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#48;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#57;&#32;&#109;&#46;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"133\" style=\"vertical-align: -15px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1275043\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1019417\">\n<p id=\"fs-id1980036\">A ball is kicked with an initial velocity of 16 m\/s in the horizontal direction and 12 m\/s in the vertical direction. (a) At what speed does the ball hit the ground? (b) For how long does the ball remain in the air? (c)What maximum height is attained by the ball?\n      <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2889503\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1790979\">\n<p id=\"fs-id1790980\">A ball is thrown horizontally from the top of a 60.0-m building and lands 100.0 m from the base of the building. Ignore air resistance. (a) How long is the ball in the air? (b) What must have been the initial horizontal component of the velocity? (c) What is the vertical component of the velocity just before the ball hits the ground? (d) What is the velocity (including both the horizontal and vertical components) of the ball just before it hits the ground?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1945253\">\n<p id=\"fs-id2167208\">(a) 3.50 s<\/p>\n<p id=\"fs-id1612943\">(b) 28.6  m\/s (c) 34.3 m\/s<\/p>\n<p id=\"fs-id916496\">(d) 44.7 m\/s,  <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-556bcaefff43050db201ea4d270eccf7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#48;&#46;&#50;&ordm;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: 0px;\" \/> below horizontal<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2197387\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2261404\">\n<p id=\"fs-id2261405\">(a) A daredevil is attempting to jump his motorcycle over a line of buses parked end to end by driving up a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ef1db3fb9b93a756e24f2b5657f8ef98_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#50;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"17\" style=\"vertical-align: 0px;\" \/> ramp at a speed of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6ec0b37ed5a2e04a289b34dc6e3dc4f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#48;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#32;&#109;&#47;&#115;&#32;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#52;&#52;&#32;&#107;&#109;&#47;&#104;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"165\" style=\"vertical-align: -4px;\" \/>. How many buses can he clear if the top of the takeoff ramp is at the same height as the bus tops and the buses are 20.0 m long? (b) Discuss what your answer implies about the margin of error in this act\u2014that is, consider how much greater the range is than the horizontal distance he must travel to miss the end of the last bus. (Neglect air resistance.)\n      <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1420192\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1078714\">\n<p id=\"fs-id1078715\">An archer shoots an arrow at a 75.0 m distant target; the bull\u2019s-eye of the target is at same height as the release height of the arrow. (a) At what angle must the arrow be released to hit the bull\u2019s-eye if its initial speed is 35.0 m\/s? In this part of the problem, explicitly show how you follow the steps involved in solving projectile motion problems. (b) There is a large tree halfway between the archer and the target with an overhanging horizontal branch 3.50 m above the release height of the arrow. Will the arrow go over or under the branch?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1970452\">\n<p id=\"fs-id1970455\">(a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a08a3439c0c4e1fab6bf553325458125_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#56;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#52;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: -1px;\" \/><\/p>\n<p id=\"fs-id1678230\">(b) The arrow will go over the branch.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1934878\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2226553\">\n<p id=\"fs-id2226554\">A rugby player passes the ball 7.00 m across the field, where it is caught at the same height as it left his hand. (a) At what angle was the ball thrown if its initial speed was 12.0 m\/s, assuming that the smaller of the two possible angles was used? (b) What other angle gives the same range, and why would it not be used? (c) How long did this pass take?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2126267\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2889922\">\n<p id=\"fs-id2889923\">Verify the ranges for the projectiles in <a href=\"#import-auto-id1904800\" class=\"autogenerated-content\">(Figure)<\/a>(a) for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-8db52593b44eaad85b9c1bad2b73df50_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#53;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"49\" style=\"vertical-align: -1px;\" \/> and the given initial velocities.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1903883\">\n<p id=\"fs-id1912801\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-350ec7850de9fe31c97382af37fd637a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#125;&#82;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#123;&#118;&#125;&#95;&#123;&#48;&#125;&#125;&#94;&#123;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#105;&#110;&#125;&#123;&#50;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#95;&#123;&#48;&#125;&#103;&#125;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#111;&#114;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#53;&ordm;&#125;&#44;&#82;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#123;&#118;&#125;&#95;&#123;&#48;&#125;&#125;&#94;&#123;&#125;&#125;&#123;&#103;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"99\" style=\"vertical-align: -116px;\" \/><\/p>\n<p id=\"fs-id1630295\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-804fe60307983ec336ecda54ca2f923c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#61;&#57;&#49;&#46;&#56;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"88\" style=\"vertical-align: -1px;\" \/> for <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-15bd02c036d13314ca146fc28c64a82f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#48;&#125;&#61;&#51;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#47;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"93\" style=\"vertical-align: -4px;\" \/>; <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-55825088a0872c0a1bae95bf771d14c1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#61;&#49;&#54;&#51;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"83\" style=\"vertical-align: -1px;\" \/><\/p>\n<p> for<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-8f5d3a9370454340d2a2e1d391237360_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#48;&#125;&#61;&#52;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#47;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"93\" style=\"vertical-align: -4px;\" \/>; <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-81e670b55779aef8c0f4aff6e6e9ffdc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#61;&#50;&#53;&#53;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"83\" style=\"vertical-align: 0px;\" \/> for <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-51724e1d544c653eecc16700055c172a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#48;&#125;&#61;&#53;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#47;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"93\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2214647\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2214182\">\n<p id=\"fs-id2214183\">Verify the ranges shown for the projectiles in <a href=\"#import-auto-id1904800\" class=\"autogenerated-content\">(Figure)<\/a>(b) for an initial velocity of 50 m\/s at the given initial angles.\n      <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2905201\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1851487\">\n<p id=\"fs-id1851488\">The cannon on a battleship can fire a shell a maximum distance of 32.0 km. (a) Calculate the initial velocity of the shell. (b) What maximum height does it reach? (At its highest, the shell is above 60% of the atmosphere\u2014but air resistance is not really negligible as assumed to make this problem easier.) (c) The ocean is not flat, because the Earth is curved. Assume that the radius of the Earth is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b9fed262f17c94d1acd1e9a669dc8ab2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#55;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#51;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"85\" style=\"vertical-align: -1px;\" \/>. How many meters lower will its surface be 32.0 km from the ship along a horizontal line parallel to the surface at the ship? Does your answer imply that error introduced by the assumption of a flat Earth in projectile motion is significant here?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2075683\">\n<p id=\"fs-id1862278\">(a) 560 m\/s<\/p>\n<p id=\"fs-id2252609\">(b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d3dee08ab28ac9984591e3d9067c3c8d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#51;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"76\" style=\"vertical-align: -1px;\" \/><\/p>\n<p id=\"fs-id2262837\">(c) 80.0 m. This error is not significant because it is only 1% of the answer in part (b).<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1925728\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2282381\">\n<p id=\"fs-id2282382\">An arrow is shot from a height of 1.5 m toward a cliff of height <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-379db1fc1f84b7ce56b92463183097f9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#72;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: 0px;\" \/>. It is shot with a velocity of 30 m\/s at an angle of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9f58822796ed6c341868803a248de619_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#48;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/> above the horizontal. It lands on the top edge of the cliff 4.0 s later. (a) What is the height of the cliff? (b) What is the maximum height reached by the arrow along its trajectory? (c) What is the arrow\u2019s impact speed just before hitting the cliff?\n      <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1745072\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2275266\">\n<p id=\"fs-id2275267\">In the standing broad jump, one squats and then pushes off with the legs to see how far one can jump. Suppose the extension of the legs from the crouch position is 0.600 m and the acceleration achieved from this position is 1.25 times the acceleration due to gravity, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d208fd391fa57c168dc0f151de829fee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/>. How far can they jump? State your assumptions. (Increased range can be achieved by swinging the arms in the direction of the jump.)<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2864553\">\n<p id=\"fs-id2864555\">1.50 m, assuming launch angle of  <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ce5319ee2d0e548b28a349b0f43ba034_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#53;&ordm;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"17\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1875777\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2864558\">\n<p id=\"fs-id2864559\">The world long jump record is 8.95 m (Mike Powell, USA, 1991). Treated as a projectile, what is the maximum range obtainable by a person if he has a take-off speed of 9.5 m\/s? State your assumptions.\n      <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2254986\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1543443\">\n<p id=\"fs-id1543444\">Serving at a speed of 170 km\/h, a tennis player hits the ball at a height of 2.5 m and an angle <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-761998727948942ceb1b5763e45f01e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/> below the horizontal. The service line is 11.9 m from the net, which is 0.91 m high. What is the angle <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-761998727948942ceb1b5763e45f01e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/> such that the ball just crosses the net? Will the ball land in the service box, whose out line is 6.40 m from the net?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id722706\">\n<p id=\"fs-id2260869\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-8b9d236b841a2f7e4254e6157f493918_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#61;&#54;&#46;&#49;&ordm;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"54\" style=\"vertical-align: -1px;\" \/><\/p>\n<p id=\"fs-id2088346\">yes, the ball lands at 5.3 m from the net<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2173828\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2088349\">\n<p id=\"fs-id2088350\">A football quarterback is moving straight backward at a speed of 2.00 m\/s when he throws a pass to a player 18.0 m straight downfield. (a) If the ball is thrown at an angle of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-aa0291ad9843d3d2d98c730ba4b76e83_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#53;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"17\" style=\"vertical-align: 0px;\" \/> relative to the ground and is caught at the same height as it is released, what is its initial speed relative to the ground? (b) How long does it take to get to the receiver? (c) What is its maximum height above its point of release?\n      <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1796436\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1979282\">\n<p id=\"fs-id1979284\">Gun sights are adjusted to aim high to compensate for the effect of gravity, effectively making the gun accurate only for a specific range. (a) If a gun is sighted to hit targets that are at the same height as the gun and 100.0 m away, how low will the bullet hit if aimed directly at a target 150.0 m away? The muzzle velocity of the bullet is 275 m\/s. (b) Discuss qualitatively how a larger muzzle velocity would affect this problem and what would be the effect of air resistance.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2175653\">\n<p id=\"fs-id2175655\">(a) \u22120.486 m<\/p>\n<p id=\"fs-id1461088\">(b) The larger the muzzle velocity, the smaller the deviation in the vertical direction, because the time of flight would be smaller. Air resistance would have the effect of decreasing the time of flight, therefore increasing the vertical deviation.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2177814\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1781566\">\n<p id=\"fs-id1781567\">An eagle is flying horizontally at a speed of 3.00 m\/s when the fish in her talons wiggles loose and falls into the lake 5.00 m below. Calculate the velocity of the fish relative to the water when it hits the water.\n      <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1914025\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2057849\">\n<p id=\"fs-id2057850\">An owl is carrying a mouse to the chicks in its nest. Its position at that time is 4.00 m west and 12.0 m above the center of the 30.0 cm diameter nest. The owl is flying east at 3.50 m\/s at an angle <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3e16ee995a39db0e4671513c4f1e853c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#48;&#46;&#48;&ordm;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"32\" style=\"vertical-align: 0px;\" \/> below the horizontal when it accidentally drops the mouse. Is the owl lucky enough to have the mouse hit the nest? To answer this question, calculate the horizontal position of the mouse when it has fallen 12.0 m.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2042074\">\n<p id=\"fs-id2042076\">4.23 m. No, the owl is not lucky; he misses the nest.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1403577\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2865734\">\n<p id=\"fs-id2865735\">Suppose a soccer player kicks the ball from a distance 30 m toward the goal. Find the initial speed of the ball if it just passes over the goal, 2.4 m above the ground, given the initial direction to be <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-aba75ce0d9b36d567c38e44174eb2fb0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#48;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"18\" style=\"vertical-align: -1px;\" \/> above the horizontal.\n      <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2260735\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1789803\">\n<p id=\"fs-id1789804\">Can a goalkeeper at her\/ his goal kick a soccer ball into the opponent\u2019s goal without the ball touching the ground? The distance will be about 95 m. A goalkeeper can give the ball a speed of 30 m\/s.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1985242\">\n<p id=\"fs-id1985244\">No, the maximum range (neglecting air resistance) is about 92 m. <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1437858\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1891285\">\n<p id=\"fs-id1891286\">The free throw line in basketball is 4.57 m (15 ft) from the basket, which is 3.05 m (10 ft) above the floor. A player standing on the free throw line throws the ball with an initial speed of 7.15 m\/s, releasing it at a height of 2.44 m (8 ft) above the floor. At what angle above the horizontal must the ball be thrown to exactly hit the basket? Note that most players will use a large initial angle rather than a flat shot because it allows for a larger margin of error. Explicitly show how you follow the steps involved in solving projectile motion problems.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1827481\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1750926\">\n<p id=\"fs-id1750928\">In 2007, Michael Carter (U.S.) set a world record in the shot put with a throw of 24.77 m. What was the initial speed of the shot if he released it at a height of 2.10 m and threw it at an angle of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a79181b720dfd7dc5feb845555576234_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#56;&#46;&#48;&ordm;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"32\" style=\"vertical-align: 0px;\" \/> above the horizontal? (Although the maximum distance for a projectile on level ground is achieved at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1142d1c44cfaf3459c45a3d6cc399899_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#53;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"17\" style=\"vertical-align: -1px;\" \/> when air resistance is neglected, the actual angle to achieve maximum range is smaller; thus, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c0fb64f1513d70c6e6b9d68d4fe0b531_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#56;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/> will give a longer range than <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1142d1c44cfaf3459c45a3d6cc399899_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#53;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"17\" style=\"vertical-align: -1px;\" \/> in the shot put.)<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1630701\">\n<p id=\"fs-id2261978\">15.0 m\/s<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1670278\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1945400\">\n<p id=\"fs-id1945401\">A basketball player is running at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c68ec733af3c4c138a7acf0fe48ec23c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#32;&#109;&#47;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"68\" style=\"vertical-align: -4px;\" \/> directly toward the basket when he jumps into the air to dunk the ball. He maintains his horizontal velocity. (a) What vertical velocity does he need to rise 0.750 m above the floor? (b) How far from the basket (measured in the horizontal direction) must he start his jump to reach his maximum height at the same time as he reaches the basket?\n          <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1779635\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1637691\">\n<p id=\"fs-id1637692\">A football player punts the ball at a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ea24dabd3bfb3eb04c56478b6c973443_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#53;&#46;&#48;&ordm;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"32\" style=\"vertical-align: -1px;\" \/> angle. Without an effect from the wind, the ball would travel 60.0 m horizontally. (a) What is the initial speed of the ball? (b) When the ball is near its maximum height it experiences a brief gust of wind that reduces its horizontal velocity by 1.50 m\/s. What distance does the ball travel horizontally?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1545352\">\n<p id=\"fs-id1545354\">(a) 24.2 m\/s\n<\/p>\n<p id=\"fs-id1796132\">(b) The ball travels a total of 57.4 m with the brief gust of wind.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2046931\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2241558\">\n<p id=\"fs-id2241559\">Prove that the trajectory of a projectile is parabolic, having the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-03029a58a61398c69613da7b4a95d1a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#120;&#125;&#43;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#98;&#120;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"99\" style=\"vertical-align: -4px;\" \/>. To obtain this expression, solve the equation <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-8ee2a5084f23a01e3053fc2fb2fa531d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#123;&#118;&#125;&#95;&#123;&#48;&#120;&#125;&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"64\" style=\"vertical-align: -3px;\" \/> for <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b4e3cbf5d4c5c6d9b702dd139f14c147_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"6\" style=\"vertical-align: 0px;\" \/> and substitute it into the expression for <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-cbca13f85783e1ef08222223be4039da_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#123;&#118;&#125;&#95;&#123;&#48;&#121;&#125;&#116;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#47;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#103;&#116;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"151\" style=\"vertical-align: -6px;\" \/> (These equations describe the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/> positions of a projectile that starts at the origin.) You should obtain an equation of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-03029a58a61398c69613da7b4a95d1a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#120;&#125;&#43;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#98;&#120;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"99\" style=\"vertical-align: -4px;\" \/> where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f56d50c26583f9a035ff6b4e3c0ca5c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"8\" style=\"vertical-align: 0px;\" \/> are constants.\n<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2133758\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2890360\">\n<p id=\"fs-id2890361\">Derive <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f494eb1a117c30bfae17a95309aa1fbd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#118;&#125;&#95;&#123;&#48;&#125;&#94;&#123;&#50;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#105;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#50;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#95;&#123;&#48;&#125;&#125;&#123;&#103;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"100\" style=\"vertical-align: -9px;\" \/> for the range of a projectile on level ground by finding the time <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b4e3cbf5d4c5c6d9b702dd139f14c147_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"6\" style=\"vertical-align: 0px;\" \/> at which <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/> becomes zero and substituting this value of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b4e3cbf5d4c5c6d9b702dd139f14c147_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"6\" style=\"vertical-align: 0px;\" \/> into the expression for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-671e8b749802e91b503a22b7972fdc2b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#123;&#120;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"49\" style=\"vertical-align: -3px;\" \/>, noting that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1538f9f2d552f517a16868a8c5def031_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#61;&#120;&#45;&#123;&#120;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"86\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"eip-id2932521\">\n<p id=\"eip-id2932523\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3cde5a4e2a3db28519546b50ede4a838_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#45;&#123;&#121;&#125;&#95;&#123;&#48;&#125;&#61;&#48;&#61;&#123;&#118;&#125;&#95;&#123;&#48;&#121;&#125;&#116;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#103;&#116;&#125;&#125;&#94;&#123;&#50;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#118;&#125;&#95;&#123;&#48;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#105;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#116;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#103;&#116;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"347\" style=\"vertical-align: -6px;\" \/>,<\/p>\n<p id=\"eip-id1990384\">so that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0b779bf03537bca420d6a7ff8e218a06_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#118;&#125;&#95;&#123;&#48;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#105;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#103;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"96\" style=\"vertical-align: -9px;\" \/><\/p>\n<p id=\"eip-id1355828\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7a0d083d63911982b4c0280f825dea89_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#123;&#120;&#125;&#95;&#123;&#48;&#125;&#61;&#123;&#118;&#125;&#95;&#123;&#48;&#120;&#125;&#116;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#118;&#125;&#95;&#123;&#48;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#111;&#115;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#116;&#61;&#82;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"249\" style=\"vertical-align: -4px;\" \/> and substituting for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b4e3cbf5d4c5c6d9b702dd139f14c147_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"6\" style=\"vertical-align: 0px;\" \/> gives:<\/p>\n<p id=\"eip-id3306953\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-928b3df0a4b85acdefba2488e141770c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#61;&#123;&#118;&#125;&#95;&#123;&#48;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#111;&#115;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#50;&#118;&#125;&#95;&#123;&#48;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#105;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#123;&#103;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#50;&#118;&#125;&#95;&#123;&#48;&#125;&#94;&#123;&#50;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#105;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#111;&#115;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#123;&#103;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"290\" style=\"vertical-align: -12px;\" \/><\/p>\n<p id=\"eip-id1599732\">since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-dd7ee0effbf92a76c89c9263e35c682e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#105;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#111;&#115;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#105;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"161\" style=\"vertical-align: -4px;\" \/> the range is:<\/p>\n<p id=\"eip-id3385487\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-df659742720fa3a551694ac61d6c06ba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#123;&#118;&#125;&#95;&#123;&#48;&#125;&#125;&#94;&#123;&#50;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#105;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#123;&#103;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"99\" style=\"vertical-align: -9px;\" \/>.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1794949\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1626931\">\n<p id=\"fs-id1626932\"><strong>Unreasonable Results<\/strong> (a) Find the maximum range of a super cannon that has a muzzle velocity of 4.0 km\/s. (b) What is unreasonable about the range you found? (c) Is the premise unreasonable or is the available equation inapplicable? Explain your answer. (d) If such a muzzle velocity could be obtained, discuss the effects of air resistance, thinning air with altitude, and the curvature of the Earth on the range of the super cannon.\n<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1815382\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2255165\">\n<p id=\"fs-id2255166\"><strong>Construct Your Own Problem<\/strong> Consider a ball tossed over a fence. Construct a problem in which you calculate the ball\u2019s needed initial velocity to just clear the fence. Among the things to determine are; the height of the fence, the distance to the fence from the point of release of the ball, and the height at which the ball is released. You should also consider whether it is possible to choose the initial speed for the ball and just calculate the angle at which it is thrown. Also examine the possibility of multiple solutions given the distances and heights you have chosen.\n <\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"glossary\" class=\"textbox shaded\">\n<h2 data-type=\"glossary-title\">Glossary<\/h2>\n<dl class=\"definition\" id=\"import-auto-id2275857\">\n<dt>air resistance<\/dt>\n<dd id=\"fs-id1668212\">a frictional force that slows the motion of objects as they travel through the air; when solving basic physics problems, air resistance is assumed to be zero<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"fs-id1405173\">\n<dt>kinematics<\/dt>\n<dd id=\"fs-id1949378\">the study of motion without regard to mass or force<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"fs-id1949381\">\n<dt>motion<\/dt>\n<dd id=\"fs-id2324535\">displacement of an object as a function of time<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id1459204\">\n<dt>projectile<\/dt>\n<dd id=\"fs-id1798529\">an object that travels through the air and experiences only acceleration due to gravity<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id1981500\">\n<dt>projectile motion<\/dt>\n<dd id=\"fs-id2253704\">the motion of an object that is subject only to the acceleration of gravity<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id2257218\">\n<dt>range<\/dt>\n<dd id=\"fs-id1883200\">the maximum horizontal distance that a projectile travels<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id2852095\">\n<dt>trajectory<\/dt>\n<dd id=\"fs-id2222621\">the path of a projectile through the air<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":211,"menu_order":1,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":"all-rights-reserved"},"chapter-type":[],"contributor":[],"license":[56],"class_list":["post-202","chapter","type-chapter","status-publish","hentry","license-all-rights-reserved"],"part":145,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/202","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/users\/211"}],"version-history":[{"count":1,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/202\/revisions"}],"predecessor-version":[{"id":203,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/202\/revisions\/203"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/parts\/145"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/202\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/media?parent=202"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapter-type?post=202"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/contributor?post=202"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/license?post=202"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}