{"id":118,"date":"2017-10-27T16:28:57","date_gmt":"2017-10-27T16:28:57","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/chapter\/falling-objects\/"},"modified":"2017-11-08T03:23:50","modified_gmt":"2017-11-08T03:23:50","slug":"falling-objects","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/chapter\/falling-objects\/","title":{"raw":"Falling Objects","rendered":"Falling Objects"},"content":{"raw":"\n<div class=\"textbox learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<ul>\n<li>Describe the effects of gravity on objects in motion.<\/li>\n<li>Describe the motion of objects that are in free fall.<\/li>\n<li>Calculate the position and velocity of objects in free fall.<\/li>\n<\/ul>\n<\/div>\n<p id=\"import-auto-id2295392\">Falling objects form an interesting class of motion problems. For example, we can estimate the depth of a vertical mine shaft by dropping a rock into it and listening for the rock to hit the bottom. By applying the kinematics developed so far to falling objects, we can examine some interesting situations and learn much about gravity in the process.<\/p>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id4178141\">\n<h1 data-type=\"title\">Gravity<\/h1>\n<p id=\"import-auto-id2006874\">The most remarkable and unexpected fact about falling objects is that, if air resistance and friction are negligible, then in a given location all objects fall toward the center of Earth with the <em data-effect=\"italics\">same constant acceleration<\/em>, <em data-effect=\"italics\">independent of their mass<\/em>. This experimentally determined fact is unexpected, because we are so accustomed to the effects of air resistance and friction that we expect light objects to fall slower than heavy ones. <\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id4126662\">\n<div class=\"bc-figcaption figcaption\">A hammer and a feather will fall with the same constant acceleration if air resistance is considered negligible. This is a general characteristic of gravity not unique to Earth, as astronaut David R. Scott demonstrated on the Moon in 1971, where the acceleration due to gravity is only [latex]1\\text{.}{\\text{67 m\/s}}^{2}[\/latex].\n      <\/div>\n<p><span data-type=\"media\" id=\"import-auto-id2553814\" data-alt=\"Positions of a feather and hammer over time as they fall on the Moon. The feather and hammer are at the exact same position at each moment in time.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_02_07_00a.jpg\" data-media-type=\"image\/jpg\" alt=\"Positions of a feather and hammer over time as they fall on the Moon. The feather and hammer are at the exact same position at each moment in time.\" width=\"300\"><\/span><\/p><\/div>\n<p id=\"import-auto-id4066514\">In the real world, air resistance can cause a lighter object to fall slower than a heavier object of the same size. A tennis ball will reach the ground after a hard baseball dropped at the same time. (It might be difficult to observe the difference if the height is not large.) Air resistance opposes the motion of an object through the air, while friction between objects\u2014such as between clothes and a laundry chute or between a stone and a pool into which it is dropped\u2014also opposes motion between them. For the ideal situations of these first few chapters, an object <em data-effect=\"italics\">falling without air resistance or friction<\/em> is defined to be in <span data-type=\"term\" id=\"import-auto-id1714641\">free-fall<\/span>.<\/p>\n<p id=\"import-auto-id4141851\">The force of gravity causes objects to fall toward the center of Earth. The acceleration of free-falling objects is therefore called the <span data-type=\"term\" id=\"import-auto-id1707599\">acceleration due to gravity<\/span>. The acceleration due to gravity is <em data-effect=\"italics\">constant<\/em>, which means we can apply the kinematics equations to any falling object where air resistance and friction are negligible. This opens a broad class of interesting situations to us. The acceleration due to gravity is so important that its magnitude is given its own symbol, [latex]g[\/latex]. It is constant at any given location on Earth and has the average value<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-636\">[latex]g=9\\text{.}{\\text{80 m\/s}}^{2}\\text{.}[\/latex]<\/div>\n<p id=\"import-auto-id3504907\">Although [latex]g[\/latex] varies from [latex]9\\text{.}{\\text{78 m\/s}}^{2}[\/latex] to [latex][\/latex][latex]9\\text{.}{\\text{83 m\/s}}^{2}[\/latex], depending on latitude, altitude, underlying geological formations, and local topography, the average value of [latex]9\\text{.}{\\text{80 m\/s}}^{2}[\/latex] will be used in this text unless otherwise specified. The direction of the acceleration due to gravity is <em data-effect=\"italics\">downward (towards the center of Earth)<\/em>. In fact, its direction <em data-effect=\"italics\">defines<\/em> what we call vertical. Note that whether the acceleration [latex]a[\/latex] in the kinematic equations has the value [latex]+g[\/latex] or [latex]-g[\/latex] depends on how we define our coordinate system. If we define the upward direction as positive, then [latex]a=-g=-9\\text{.}{\\text{80 m\/s}}^{2}[\/latex], and if we define the downward direction as positive, then [latex]a=g=9\\text{.}{\\text{80 m\/s}}^{2}[\/latex].<\/p>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id2854808\">\n<h1 data-type=\"title\">One-Dimensional Motion Involving Gravity<\/h1>\n<p id=\"import-auto-id1354046\">The best way to see the basic features of motion involving gravity is to start with the simplest situations and then progress toward more complex ones. So we start by considering straight up and down motion with no air resistance or friction. These assumptions mean that the velocity (if there is any) is vertical. If the object is dropped, we know the initial velocity is zero. Once the object has left contact with whatever held or threw it, the object is in free-fall. Under these circumstances, the motion is one-dimensional and has constant acceleration of magnitude [latex]g[\/latex]. We will also represent vertical displacement with the symbol [latex]y[\/latex] and use [latex]x[\/latex] for horizontal displacement.<\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id1931991\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Kinematic Equations for Objects in Free-Fall where Acceleration = -<em data-effect=\"italics\">g<\/em><\/div>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2222965\">[latex]v={v}_{0}-\\text{gt}[\/latex]<\/div>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id4048491\">[latex]y={y}_{0}+{v}_{0}t-\\frac{1}{2}{\\text{gt}}^{2}[\/latex]<\/div>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id4019890\">[latex]{v}^{2}={v}_{0}^{2}-2g\\left(y-{y}_{0}\\right)[\/latex]<\/div>\n<\/div>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id4067058\">\n<div data-type=\"title\" class=\"title\">Calculating Position and Velocity of a Falling Object: A Rock Thrown Upward<\/div>\n<p id=\"import-auto-id4149723\">A person standing on the edge of a high cliff throws a rock straight up with an initial velocity of 13.0 m\/s<em data-effect=\"italics\">.<\/em> The rock misses the edge of the cliff as it falls back to earth. Calculate the position and velocity of the rock 1.00 s, 2.00 s, and 3.00 s after it is thrown, neglecting the effects of air resistance.<\/p>\n<p id=\"import-auto-id4127381\"><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id1451572\">Draw a sketch.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id2188586\"><span data-type=\"media\" id=\"import-auto-id3561557\" data-alt=\"Velocity vector arrow pointing up in the positive y direction, labeled v sub 0 equals thirteen point 0 meters per second. Acceleration vector arrow pointing down in the negative y direction, labeled a equals negative 9 point 8 meters per second squared.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_02_07_00b.jpg\" data-media-type=\"image\/jpg\" alt=\"Velocity vector arrow pointing up in the positive y direction, labeled v sub 0 equals thirteen point 0 meters per second. Acceleration vector arrow pointing down in the negative y direction, labeled a equals negative 9 point 8 meters per second squared.\" width=\"350\"><\/span><\/div>\n<p id=\"import-auto-id1345513\">We are asked to determine the position [latex]y[\/latex] at various times. It is reasonable to take the initial position [latex]{y}_{0}[\/latex] to be zero. This problem involves one-dimensional motion in the vertical direction. We use plus and minus signs to indicate direction, with up being positive and down negative. Since up is positive, and the rock is thrown upward, the initial velocity must be positive too. The acceleration due to gravity is downward, so [latex]a[\/latex] is negative. It is crucial that the initial velocity and the acceleration due to gravity have opposite signs. Opposite signs indicate that the acceleration due to gravity opposes the initial motion and will slow and eventually reverse it. <\/p>\n<p id=\"import-auto-id4095755\">Since we are asked for values of position and velocity at three times, we will refer to these as [latex]{y}_{1}[\/latex] and [latex]{v}_{1}[\/latex]; <em data-effect=\"italics\">[latex]{y}_{2}[\/latex]<\/em> and [latex]{v}_{2}[\/latex]; and [latex]{y}_{3}[\/latex] and [latex]{v}_{3}[\/latex].<\/p>\n<p id=\"import-auto-id2217851\"><strong>Solution for Position <\/strong>[latex]{y}_{1}[\/latex]<\/p>\n<p id=\"import-auto-id1659278\">1. Identify the knowns. We know that [latex]{y}_{0}=0[\/latex]; [latex]{v}_{0}=\\text{13}\\text{.}\\text{0 m\/s}[\/latex]; [latex]a=-g=-9\\text{.}{\\text{80 m\/s}}^{2}[\/latex]; and [latex]t=1\\text{.}\\text{00 s}[\/latex]. <\/p>\n<p id=\"import-auto-id3627669\">2. Identify the best equation to use. We will use [latex]y={y}_{0}+{v}_{0}t+\\frac{1}{2}{\\text{at}}^{2}[\/latex] because it includes only one unknown, [latex]y[\/latex] (or [latex]{y}_{1}[\/latex], here),<em data-effect=\"italics\"> which is the value we want to find.<\/em><\/p>\n<p id=\"import-auto-id3602200\">3. Plug in the known values and solve for [latex]{y}_{1}[\/latex].<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2302522\">[latex]y{}_{1}\\text{}=0+\\left(\\text{13}\\text{.}\\text{0 m\/s}\\right)\\left(1\\text{.}\\text{00 s}\\right)+\\frac{1}{2}\\left(-9\\text{.}\\text{80}\\phantom{\\rule{0.25em}{0ex}}{\\text{m\/s}}^{2}\\right){\\left(1\\text{.}\\text{00 s}\\right)}^{2}=8\\text{.}\\text{10}\\phantom{\\rule{0.25em}{0ex}}\\text{m}[\/latex]<\/div>\n<p id=\"import-auto-id1772102\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id2111072\">The rock is 8.10 m above its starting point at [latex]t=1\\text{.}\\text{00}[\/latex] s, since [latex]{y}_{1}&gt;{y}_{0}[\/latex]. It could be <em data-effect=\"italics\">moving<\/em> up or down; the only way to tell is to calculate [latex]{v}_{1}[\/latex] and find out if it is positive or negative. <\/p>\n<p id=\"import-auto-id2223740\"><strong>Solution for Velocity <\/strong>[latex]{v}_{1}[\/latex]<\/p>\n<p id=\"import-auto-id4138785\">1. Identify the knowns. We know that [latex]{y}_{0}=0[\/latex]; [latex]{v}_{0}=\\text{13}\\text{.}\\text{0 m\/s}[\/latex]; [latex]a=-g=-9\\text{.}{\\text{80 m\/s}}^{2}[\/latex]; and [latex]t=1\\text{.}\\text{00 s}[\/latex]. We also know from the solution above that [latex]{y}_{1}=8\\text{.}\\text{10 m}[\/latex].<\/p>\n<p id=\"import-auto-id4138788\">2. Identify the best equation to use. The most straightforward is [latex]v={v}_{0}-\\text{gt}[\/latex] (from [latex]v={v}_{0}+\\text{at}[\/latex], where [latex]a=\\text{gravitational acceleration}=-g[\/latex]).<\/p>\n<p id=\"import-auto-id1688801\">3. Plug in the knowns and solve. <\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id1688803\">[latex]{v}_{1}={v}_{0}-\\text{gt}=\\text{13}\\text{.}\\text{0 m\/s}-\\left(9\\text{.}{\\text{80 m\/s}}^{2}\\right)\\left(1\\text{.}\\text{00 s}\\right)=3\\text{.}\\text{20 m\/s}[\/latex]<\/div>\n<p id=\"import-auto-id2025041\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id3524154\">The positive value for [latex]{v}_{1}[\/latex] means that the rock is still heading upward at [latex]t=1\\text{.}\\text{00}\\phantom{\\rule{0.25em}{0ex}}\\text{s}[\/latex]. However, it has slowed from its original 13.0<em data-effect=\"italics\"> m\/s, as expected.<\/em><\/p>\n<p id=\"import-auto-id2034327\"><strong>Solution for Remaining Times<\/strong><\/p>\n<p id=\"import-auto-id2001485\">The procedures for calculating the position and velocity at [latex]t=2\\text{.}\\text{00}\\phantom{\\rule{0.25em}{0ex}}\\text{s}[\/latex] and [latex]3\\text{.}\\text{00 s}[\/latex] are the same as those above. The results are summarized in <a href=\"#eip-304\" class=\"autogenerated-content\">(Figure)<\/a> and illustrated in <a href=\"#import-auto-id4064055\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<table id=\"eip-304\" summary=\"Table with four columns showing the time, position, velocity, and acceleration of a rock thrown in the air. Times are listed in column one, and corresponding position, velocity, and acceleration are listed in the next three columns.\">\n<caption><span data-type=\"title\">Results<\/span><\/caption>\n<thead>\n<tr>\n<th data-align=\"center\">\n            Time, <em data-effect=\"italics\">t<\/em>\n            <\/th>\n<th data-align=\"center\">\n              Position, <em data-effect=\"italics\">y<\/em>\n            <\/th>\n<th data-align=\"center\">\n              Velocity, <em data-effect=\"italics\">v<\/em>\n            <\/th>\n<th data-align=\"center\">\n              Acceleration, <em data-effect=\"italics\">a<\/em>\n            <\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td data-align=\"center\">[latex]1\\text{.}\\text{00 s}[\/latex]<\/td>\n<td data-align=\"center\">[latex]8\\text{.}\\text{10 m}[\/latex]<\/td>\n<td data-align=\"center\">[latex]3\\text{.}\\text{20 m\/s}[\/latex]<\/td>\n<td data-align=\"center\">[latex]-9\\text{.}{\\text{80 m\/s}}^{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">[latex]2\\text{.}\\text{00 s}[\/latex]<\/td>\n<td data-align=\"center\">[latex]6\\text{.}\\text{40 m}[\/latex]<\/td>\n<td data-align=\"center\">[latex]-6\\text{.}\\text{60 m\/s}[\/latex]<\/td>\n<td data-align=\"center\">[latex]-9\\text{.}{\\text{80 m\/s}}^{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">[latex]3\\text{.}\\text{00 s}[\/latex]<\/td>\n<td data-align=\"center\">[latex]-5\\text{.}\\text{10 m}[\/latex]<\/td>\n<td data-align=\"center\">[latex]-\\text{16}\\text{.}\\text{4 m\/s}[\/latex]<\/td>\n<td data-align=\"center\">[latex]-9\\text{.}{\\text{80 m\/s}}^{2}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"import-auto-id3537673\">Graphing the data helps us understand it more clearly.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id4064055\">\n<div class=\"bc-figcaption figcaption\">Vertical position, vertical velocity, and vertical acceleration vs. time for a rock thrown vertically up at the edge of a cliff. Notice that velocity changes linearly with time and that acceleration is constant. <em data-effect=\"italics\">Misconception Alert!<\/em> Notice that the position vs. time graph shows vertical position only. It is easy to get the impression that the graph shows some horizontal motion\u2014the shape of the graph looks like the path of a projectile. But this is not the case; the horizontal axis is <em data-effect=\"italics\">time<\/em>, not space. The actual path of the rock in space is straight up, and straight down.\n      <\/div>\n<p><span data-type=\"media\" id=\"import-auto-id4064056\" data-alt=\"Three panels showing three graphs. The top panel shows a graph of vertical position in meters versus time in seconds. The line begins at the origin and has a positive slope that decreases over time until it hits a turning point between seconds 1 and 2. After that it has a negative slope that increases over time. The middle panel shows a graph of velocity in meters per second versus time in seconds. The line is straight, with a negative slope, beginning at time zero velocity of thirteen meters per second and ending at time 3 seconds with a velocity just over negative sixteen meters per second. The bottom panel shows a graph of acceleration in meters per second squared versus time in seconds. The line is straight and flat at a y value of negative 9 point 80 meters per second squared from time 0 to time 3 seconds.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_02_06_01.jpg\" data-media-type=\"image\/jpg\" alt=\"Three panels showing three graphs. The top panel shows a graph of vertical position in meters versus time in seconds. The line begins at the origin and has a positive slope that decreases over time until it hits a turning point between seconds 1 and 2. After that it has a negative slope that increases over time. The middle panel shows a graph of velocity in meters per second versus time in seconds. The line is straight, with a negative slope, beginning at time zero velocity of thirteen meters per second and ending at time 3 seconds with a velocity just over negative sixteen meters per second. The bottom panel shows a graph of acceleration in meters per second squared versus time in seconds. The line is straight and flat at a y value of negative 9 point 80 meters per second squared from time 0 to time 3 seconds.\" width=\"300\"><\/span><\/p><\/div>\n<p id=\"import-auto-id1681385\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id2220081\">The interpretation of these results is important. At 1.00 s the rock is above its starting point and heading upward, since [latex]{y}_{1}[\/latex] and [latex]{v}_{1}[\/latex] are both positive. At 2.00 s, the rock is still above its starting point, but the negative velocity means it is moving downward. At 3.00 s, both [latex]{y}_{3}[\/latex] and [latex]{v}_{3}[\/latex] are negative, meaning the rock is below its starting point and continuing to move downward. Notice that when the rock is at its highest point (at 1.5 s), its velocity is zero, but its acceleration is still [latex]-9\\text{.}{\\text{80 m\/s}}^{2}[\/latex]. Its acceleration is [latex]-9\\text{.}{\\text{80 m\/s}}^{2}[\/latex]<sup> for the whole trip\u2014while it is moving up and while it is moving down. Note that the values for [latex]y[\/latex]<em data-effect=\"italics\"> are the positions (or displacements) of the rock, not the total distances traveled. Finally, note that free-fall applies to upward motion as well as downward. Both have the same acceleration\u2014the acceleration due to gravity, which remains constant the entire time. Astronauts training in the famous Vomit Comet, for example, experience free-fall while arcing up as well as down, as we will discuss in more detail later.<\/em><\/sup><\/p>\n<\/div>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id1773192\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Making Connections: Take-Home Experiment\u2014Reaction Time<\/div>\n<p id=\"import-auto-id2293959\">A simple experiment can be done to determine your reaction time. Have a friend hold a ruler between your thumb and index finger, separated by about 1 cm. Note the mark on the ruler that is right between your fingers. Have your friend drop the ruler unexpectedly, and try to catch it between your two fingers. Note the new reading on the ruler. Assuming acceleration is that due to gravity, calculate your reaction time. How far would you travel in a car (moving at 30 m\/s) if the time it took your foot to go from the gas pedal to the brake was twice this reaction time?<\/p>\n<\/div>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id2186600\">\n<div data-type=\"title\" class=\"title\">Calculating Velocity of a Falling Object: A Rock Thrown Down<\/div>\n<p id=\"import-auto-id4097406\">What happens if the person on the cliff throws the rock straight down, instead of straight up? To explore this question, calculate the velocity of the rock when it is 5.10 m below the starting point, and has been thrown downward with an initial speed of 13.0 m\/s.<\/p>\n<p id=\"import-auto-id4097414\"><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id2150745\">Draw a sketch. <\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id2150750\"><span data-type=\"media\" id=\"import-auto-id2150751\" data-alt=\"Velocity vector arrow pointing down in the negative y direction and labeled v sub zero equals negative thirteen point 0 meters per second. Acceleration vector arrow also pointing down in the negative y direction, labeled a equals negative 9 point 80 meters per second squared.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_02_06_01a.jpg\" data-media-type=\"image\/jpg\" alt=\"Velocity vector arrow pointing down in the negative y direction and labeled v sub zero equals negative thirteen point 0 meters per second. Acceleration vector arrow also pointing down in the negative y direction, labeled a equals negative 9 point 80 meters per second squared.\" width=\"350\"><\/span><\/div>\n<p id=\"import-auto-id3604721\">Since up is positive, the final position of the rock will be negative because it finishes below the starting point at [latex]{y}_{0}=0[\/latex]. Similarly, the initial velocity is downward and therefore negative, as is the acceleration due to gravity. We expect the final velocity to be negative since the rock will continue to move downward.<\/p>\n<p id=\"import-auto-id4110111\"><strong>Solution<\/strong><\/p>\n<p id=\"import-auto-id1788409\">1. Identify the knowns. [latex]{y}_{0}=0[\/latex];<br>\n[latex]{y}_{1}=-5\\text{.}\\text{10 m}[\/latex];<br>\n[latex]{v}_{0}=-\\text{13}\\text{.0 m\/s}[\/latex];<br>\n[latex]a=-g=-9\\text{.}\\text{80 m}{\\text{\/s}}^{2}[\/latex].<\/p>\n<p id=\"import-auto-id1782245\">2. Choose the kinematic equation that makes it easiest to solve the problem. The equation [latex]{v}^{2}={v}_{0}^{2}+2a\\left(y-{y}_{0}\\right)[\/latex] works well because the only unknown in it is [latex]v[\/latex]. (We will plug [latex]{y}_{1}[\/latex] in for [latex]y[\/latex].)<\/p>\n<p id=\"import-auto-id2025015\">3. Enter the known values<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2025017\">[latex]{v}^{2}={\\left(-\\text{13}\\text{.}\\text{0 m\/s}\\right)}^{2}+2\\left(-9\\text{.}{\\text{80 m\/s}}^{2}\\right)\\left(-5\\text{.}\\text{10 m}-\\text{0 m}\\right)=\\text{268}\\text{.}{\\text{96 m}}^{2}{\\text{\/s}}^{2},[\/latex]<\/div>\n<p id=\"import-auto-id1763708\">where we have retained extra significant figures because this is an intermediate result.<\/p>\n<p id=\"import-auto-id1763705\">Taking the square root, and noting that a square root can be positive or negative, gives<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id1763704\">[latex]v=\u00b1\\text{16}\\text{.4 m\/s}.[\/latex]<\/div>\n<p id=\"import-auto-id1797916\">The negative root is chosen to indicate that the rock is still heading down. Thus,<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2563747\">[latex]v=-\\text{16}\\text{.4 m\/s}.[\/latex]<\/div>\n<p id=\"import-auto-id2561834\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id2563753\">Note that <em data-effect=\"italics\">this is exactly the same velocity the rock had at this position when it was thrown straight upward with the same initial speed<\/em>. (See <a href=\"#fs-id4067058\" class=\"autogenerated-content\">(Figure)<\/a> and <a href=\"#import-auto-id4173440\" class=\"autogenerated-content\">(Figure)<\/a>(a).) This is not a coincidental result. Because we only consider the acceleration due to gravity in this problem, the <em data-effect=\"italics\">speed<\/em> of a falling object depends only on its initial speed and its vertical position relative to the starting point. For example, if the velocity of the rock is calculated at a height of 8.10 m above the starting point (using the method from <a href=\"#fs-id4067058\" class=\"autogenerated-content\">(Figure)<\/a>) when the initial velocity is 13.0 m\/s straight up, a result of [latex]\u00b13\\text{.}\\text{20 m\/s}[\/latex] is obtained. Here both signs are meaningful; the positive value occurs when the rock is at 8.10 m and heading up, and the negative value occurs when the rock is at 8.10 m and heading back down. It has the same <em data-effect=\"italics\">speed<\/em> but the opposite direction. <\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id4173440\">\n<div class=\"bc-figcaption figcaption\">(a) A person throws a rock straight up, as explored in <a href=\"#fs-id4067058\" class=\"autogenerated-content\">(Figure)<\/a>. The arrows are velocity vectors at 0, 1.00, 2.00, and 3.00 s. (b) A person throws a rock straight down from a cliff with the same initial speed as before, as in <a href=\"#fs-id2186600\" class=\"autogenerated-content\">(Figure)<\/a>. Note that at the same distance below the point of release, the rock has the same velocity in both cases. <\/div>\n<p><span data-type=\"media\" id=\"import-auto-id4173441\" data-alt=\"Two figures are shown. At left, a man standing on the edge of a cliff throws a rock straight up with an initial speed of thirteen meters per second. At right, the man throws the rock straight down with a speed of thirteen meters per second. In both figures, a line indicates the rock\u2019s trajectory. When the rock is thrown straight up, it has a speed of minus sixteen point four meters per second at minus five point one zero meters below the point where the man released the rock. When the rock is thrown straight down, the velocity is the same at this position.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_02_06_00b.jpg\" data-media-type=\"image\/jpg\" alt=\"Two figures are shown. At left, a man standing on the edge of a cliff throws a rock straight up with an initial speed of thirteen meters per second. At right, the man throws the rock straight down with a speed of thirteen meters per second. In both figures, a line indicates the rock\u2019s trajectory. When the rock is thrown straight up, it has a speed of minus sixteen point four meters per second at minus five point one zero meters below the point where the man released the rock. When the rock is thrown straight down, the velocity is the same at this position.\" width=\"450\"><\/span><\/p><\/div>\n<p id=\"import-auto-id4180408\">Another way to look at it is this: In <a href=\"#fs-id4067058\" class=\"autogenerated-content\">(Figure)<\/a>, the rock is thrown up with an initial velocity of [latex]\\text{13}\\text{.0 m\/s}[\/latex]. It rises and then falls back down. When its position is [latex]y=0[\/latex] on its way back down, its velocity is [latex]-\\text{13}\\text{.0 m\/s}[\/latex]. That is, it has the same speed on its way down as on its way up. We would then expect its velocity at a position of [latex]y=-5\\text{.}\\text{10 m}[\/latex] to be the same whether we have thrown it upwards at [latex]+\\text{13}\\text{.0 m\/s}[\/latex] or thrown it downwards at [latex]-\\text{13}\\text{.0 m\/s}[\/latex]. The velocity of the rock on its way down from [latex]y=0[\/latex] is the same whether we have thrown it up or down to start with, as long as the speed with which it was initially thrown is the same. <\/p>\n<\/div>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id3526422\">\n<div data-type=\"title\" class=\"title\">Find <em data-effect=\"italics\">g<\/em> from Data on a Falling Object<\/div>\n<p id=\"import-auto-id2585093\">The acceleration due to gravity on Earth differs slightly from place to place, depending on topography (e.g., whether you are on a hill or in a valley) and subsurface geology (whether there is dense rock like iron ore as opposed to light rock like salt beneath you.) The precise acceleration due to gravity can be calculated from data taken in an introductory physics laboratory course. An object, usually a metal ball for which air resistance is negligible, is dropped and the time it takes to fall a known distance is measured. See, for example, <a href=\"#import-auto-id4097254\" class=\"autogenerated-content\">(Figure)<\/a>. Very precise results can be produced with this method if sufficient care is taken in measuring the distance fallen and the elapsed time. <\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id4097254\">\n<div class=\"bc-figcaption figcaption\">Positions and velocities of a metal ball released from rest when air resistance is negligible. Velocity is seen to increase linearly with time while displacement increases with time squared. Acceleration is a constant and is equal to gravitational acceleration.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id4097255\" data-alt=\"Figure has four panels. The first panel (on the top) is an illustration of a ball falling toward the ground at intervals of one tenth of a second. The space between the vertical position of the ball at one time step and the next increases with each time step. At time equals 0, position and velocity are also 0. At time equals 0 point 1 seconds, y position equals negative 0 point 049 meters and velocity is negative 0 point 98 meters per second. At 0 point 5 seconds, y position is negative 1 point 225 meters and velocity is negative 4 point 90 meters per second. The second panel (in the middle) is a line graph of position in meters versus time in seconds. Line begins at the origin and slopes down with increasingly negative slope. The third panel (bottom left) is a line graph of velocity in meters per second versus time in seconds. Line is straight, beginning at the origin and with a constant negative slope. The fourth panel (bottom right) is a line graph of acceleration in meters per second squared versus time in seconds. Line is flat, at a constant y value of negative 9 point 80 meters per second squared.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_02_06_02.jpg\" data-media-type=\"image\/jpg\" alt=\"Figure has four panels. The first panel (on the top) is an illustration of a ball falling toward the ground at intervals of one tenth of a second. The space between the vertical position of the ball at one time step and the next increases with each time step. At time equals 0, position and velocity are also 0. At time equals 0 point 1 seconds, y position equals negative 0 point 049 meters and velocity is negative 0 point 98 meters per second. At 0 point 5 seconds, y position is negative 1 point 225 meters and velocity is negative 4 point 90 meters per second. The second panel (in the middle) is a line graph of position in meters versus time in seconds. Line begins at the origin and slopes down with increasingly negative slope. The third panel (bottom left) is a line graph of velocity in meters per second versus time in seconds. Line is straight, beginning at the origin and with a constant negative slope. The fourth panel (bottom right) is a line graph of acceleration in meters per second squared versus time in seconds. Line is flat, at a constant y value of negative 9 point 80 meters per second squared.\" width=\"550\"><\/span><\/p><\/div>\n<p id=\"import-auto-id4146639\">Suppose the ball falls 1.0000 m in 0.45173 s. Assuming the ball is not affected by air resistance, what is the precise acceleration due to gravity at this location?<\/p>\n<p id=\"import-auto-id4146642\"><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id4051154\">Draw a sketch. <\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id4051158\"><span data-type=\"media\" id=\"import-auto-id4051159\" data-alt=\"The figure shows a green dot labeled v sub zero equals zero meters per second, a purple downward pointing arrow labeled a equals question mark, and an x y coordinate system with the y axis pointing vertically up and the x axis pointing horizontally to the right.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_02_06_02b.jpg\" data-media-type=\"image\/jpg\" alt=\"The figure shows a green dot labeled v sub zero equals zero meters per second, a purple downward pointing arrow labeled a equals question mark, and an x y coordinate system with the y axis pointing vertically up and the x axis pointing horizontally to the right.\" width=\"350\"><\/span><\/div>\n<p id=\"import-auto-id4025931\">We need to solve for acceleration [latex]a[\/latex]. Note that in this case, displacement is downward and therefore negative, as is acceleration. <\/p>\n<p id=\"import-auto-id1551030\"><strong>Solution<\/strong><\/p>\n<p id=\"import-auto-id1551034\">1. Identify the knowns. [latex]{y}_{0}=0[\/latex];<br>\n[latex]y=\u20131\\text{.0000 m}[\/latex];<br>\n[latex]t=0\\text{.45173}[\/latex]; [latex]{v}_{0}=0[\/latex]. <\/p>\n<p id=\"import-auto-id4043850\">2. Choose the equation that allows you to solve for [latex]a[\/latex] using the known values.<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id3578358\">[latex]y={y}_{0}+{v}_{0}t+\\frac{1}{2}{\\text{at}}^{2}[\/latex]<\/div>\n<p id=\"import-auto-id4160974\">3. Substitute 0 for [latex]{v}_{0}[\/latex] and rearrange the equation to solve for [latex]a[\/latex]. Substituting 0 for [latex]{v}_{0}[\/latex] yields<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id3538274\">[latex]y={y}_{0}+\\frac{1}{2}{\\text{at}}^{2}\\text{.}[\/latex]<\/div>\n<p id=\"import-auto-id3504490\">Solving for [latex]a[\/latex] gives<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id3504484\">[latex]a=\\frac{2\\left(y-{y}_{0}\\right)}{{t}^{2}}\\text{.}[\/latex]<\/div>\n<p id=\"import-auto-id4128814\">4. Substitute known values yields<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2571438\">[latex]a=\\frac{2\\left(-1\\text{.}\\text{0000 m \u2013 0}\\right)}{\\left(0\\text{.}\\text{45173 s}{\\right)}^{2}}=-9\\text{.}{\\text{8010 m\/s}}^{2},[\/latex]<\/div>\n<p id=\"import-auto-id4129554\">so, because [latex]a=-g[\/latex] with the directions we have chosen,<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id1706716\">[latex]g=9\\text{.}{\\text{8010 m\/s}}^{2}.[\/latex]<\/div>\n<p id=\"import-auto-id2018012\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id2007329\">The negative value for [latex]a[\/latex] indicates that the gravitational acceleration is downward, as expected. We expect the value to be somewhere around the average value of [latex]9\\text{.}{\\text{80 m\/s}}^{2}[\/latex], so [latex]9\\text{.}{\\text{8010 m\/s}}^{2}[\/latex] makes sense. Since the data going into the calculation are relatively precise, this value for [latex]g[\/latex] is more precise than the average value of [latex]9\\text{.}{\\text{80 m\/s}}^{2}[\/latex]; it represents the local value for the acceleration due to gravity.<\/p>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id4172780\" data-element-type=\"check-understanding\" data-label=\"\">\n<div data-type=\"title\">Check Your Understanding<\/div>\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1747475\">\n<p id=\"import-auto-id2325444\">A chunk of ice breaks off a glacier and falls 30.0 meters before it hits the water. Assuming it falls freely (there is no air resistance), how long does it take to hit the water?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1652531\">\n<p id=\"import-auto-id2358780\">We know that initial position [latex]{y}_{0}=0[\/latex], final position [latex]y=\\text{\u221230}\\text{.}\\text{0 m}[\/latex], and [latex]a=-g=-9\\text{.}{\\text{80 m\/s}}^{2}[\/latex]. We can then use the equation [latex]y={y}_{0}+{v}_{0}t+\\frac{1}{2}{\\text{at}}^{2}[\/latex] to solve for [latex]t[\/latex]. Inserting [latex]a=-g[\/latex], we obtain<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2358783\">[latex]\\begin{array}{lll}y&amp; =&amp; 0+0-\\frac{1}{2}{\\text{gt}}^{2}\\\\ {t}^{2}&amp; =&amp; \\frac{2y}{-g}\\\\ t&amp; =&amp; \u00b1\\sqrt{\\frac{2y}{-g}}=\u00b1\\sqrt{\\frac{2\\left(-\\text{30.0 m}\\right)}{-9.80 m{\\text{\/s}}^{2}}}=\u00b1\\sqrt{\\text{6.12}\\phantom{\\rule{0.25em}{0ex}}{s}^{2}}=\\text{2.47 s}\\approx \\text{2.5 s}\\end{array}[\/latex]<\/div>\n<p id=\"import-auto-id2307769\">where we take the positive value as the physically relevant answer. Thus, it takes about 2.5 seconds for the piece of ice to hit the water.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id2006949\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">PhET Explorations: Equation Grapher<\/div>\n<p id=\"import-auto-id2044902\">Learn about graphing polynomials. The shape of the curve changes as the constants are adjusted. View the curves for the individual terms (e.g. [latex]y=\\text{bx}[\/latex]) to see how they add to generate the polynomial curve.<\/p>\n<div class=\"bc-figure figure\" id=\"eip-id2087188\">\n<div class=\"bc-figcaption figcaption\"><a href=\"\/resources\/e6ee9717e3c9be4b4c32e56fa95242ccc73d2262\/equation-grapher_en.jar\">Equation Grapher<\/a><\/div>\n<p><span data-type=\"media\" id=\"Phet_module_2.7\" data-alt=\"\"><a href=\"\/resources\/e6ee9717e3c9be4b4c32e56fa95242ccc73d2262\/equation-grapher_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>\n<div class=\"section-summary\" data-depth=\"1\" id=\"fs-id1822906\">\n<h1 data-type=\"title\">Section Summary<\/h1>\n<ul id=\"fs-id3611205\">\n<li id=\"import-auto-id1715211\">An object in free-fall experiences constant acceleration if air resistance is negligible.<\/li>\n<li id=\"import-auto-id1715213\">On Earth, all free-falling objects have an acceleration due to gravity [latex]g[\/latex], which averages\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id3547826\">[latex]g=9\\text{.}{\\text{80 m\/s}}^{2}.[\/latex]<\/div>\n<\/li>\n<li id=\"import-auto-id2150922\">Whether the acceleration <em data-effect=\"italics\">a <\/em>should be taken as [latex]+g[\/latex] or [latex]-g[\/latex] is determined by your choice of coordinate system. If you choose the upward direction as positive, [latex]a=-g=-9\\text{.}\\text{80 m}{\\text{\/s}}^{2}[\/latex] is negative. In the opposite case, [latex]a=\\mathrm{+g}=9\\text{.}{\\text{80 m\/s}}^{2}[\/latex] is positive. Since acceleration is constant, the kinematic equations above can be applied with the appropriate [latex]+g[\/latex] or<br>\n[latex]-g[\/latex] substituted for [latex]a[\/latex].<\/li>\n<li id=\"import-auto-id4051701\">For objects in free-fall, up is normally taken as positive for displacement, velocity, and acceleration.<\/li>\n<\/ul>\n<\/div>\n<div class=\"conceptual-questions\" data-depth=\"1\" id=\"fs-id1358164\" data-element-type=\"conceptual-questions\">\n<h1 data-type=\"title\">Conceptual Questions<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1427339\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id4095246\">\n<p id=\"import-auto-id2271017\">What is the acceleration of a rock thrown straight upward on the way up? At the top of its flight? On the way down?  <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3606158\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1713184\">\n<p id=\"import-auto-id2271030\">An object that is thrown straight up falls back to Earth. This is one-dimensional motion. (a) When is its velocity zero? (b) Does its velocity change direction? (c) Does the acceleration due to gravity have the same sign on the way up as on the way down? <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2044867\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2006404\">\n<p id=\"import-auto-id2281895\">Suppose you throw a rock nearly straight up at a coconut in a palm tree, and the rock misses on the way up but hits the coconut on the way down. Neglecting air resistance, how does the speed of the rock when it hits the coconut on the way down compare with what it would have been if it had hit the coconut on the way up? Is it more likely to dislodge the coconut on the way up or down? Explain. <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1773324\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2300211\">\n<p id=\"import-auto-id3728320\">If an object is thrown straight up and air resistance is negligible, then its speed when it returns to the starting point is the same as when it was released. If air resistance were not negligible, how would its speed upon return compare with its initial speed? How would the maximum height to which it rises be affected? <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1776230\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2158441\">\n<p id=\"import-auto-id3639568\">The severity of a fall depends on your speed when you strike the ground. All factors but the acceleration due to gravity being the same, how many times higher could a safe fall on the Moon be than on Earth (gravitational acceleration on the Moon is about 1\/6 that of the Earth)? <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2271654\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id4018154\">\n<p id=\"import-auto-id3547965\">How many times higher could an astronaut jump on the Moon than on Earth if his takeoff speed is the same in both locations (gravitational acceleration on the Moon is about 1\/6 of [latex]g[\/latex] on Earth)? <\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"problems-exercises\" data-depth=\"1\" id=\"fs-id3514521\" data-element-type=\"problems-exercises\">\n<h1 data-type=\"title\">Problems &amp; Exercises<\/h1>\n<p id=\"eip-150\">Assume air resistance is negligible unless otherwise stated.<\/p>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1516821\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2558696\">\n<p id=\"import-auto-id1658423\">Calculate the displacement and velocity at times of (a) 0.500, (b) 1.00, (c) 1.50, and (d) 2.00 s for a ball thrown straight up with an initial velocity of 15.0 m\/s. Take the point of release to be [latex]{y}_{0}=0[\/latex].<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1722520\">\n<p id=\"import-auto-id2303239\">(a) [latex]{y}_{1}=6\\text{.}\\text{28 m}[\/latex]; [latex]{v}_{1}=\\text{10}\\text{.}\\text{1 m\/s}[\/latex]<\/p>\n<p id=\"import-auto-id1725030\">(b) [latex]{y}_{2}=\\text{10}\\text{.}\\text{1 m}[\/latex]; [latex]{v}_{2}=5\\text{.}\\text{20 m\/s}[\/latex]<\/p>\n<p id=\"import-auto-id1658704\">(c) [latex]{y}_{3}=11\\text{.}5 m[\/latex]; [latex]{v}_{3}=0\\text{.300 m\/s}[\/latex]<\/p>\n<p id=\"import-auto-id1725057\">(d) [latex]{y}_{4}=10\\text{.4 m}[\/latex]; [latex]{v}_{4}=-4\\text{.60 m\/s}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1746555\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id4057791\">\n<p id=\"import-auto-id3728159\">Calculate the displacement and velocity at times of (a) 0.500, (b) 1.00, (c) 1.50, (d) 2.00, and (e) 2.50 s for a rock thrown straight down with an initial velocity of 14.0 m\/s from the Verrazano Narrows Bridge in New York City. The roadway of this bridge is 70.0 m above the water.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1781525\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1781526\">\n<p id=\"import-auto-id2354922\">A basketball referee tosses the ball straight up for the starting tip-off. At what velocity must a basketball player leave the ground to rise 1.25 m above the floor in an attempt to get the ball?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2300377\">\n<p id=\"import-auto-id2354930\">[latex]{v}_{0}=4\\text{.}\\text{95 m\/s}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1582773\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2593341\">\n<p id=\"import-auto-id1590258\">A rescue helicopter is hovering over a person whose boat has sunk. One of the rescuers throws a life preserver straight down to the victim with an initial velocity of 1.40 m\/s and observes that it takes 1.8 s to reach the water. (a) List the knowns in this problem. (b) How high above the water was the preserver released? Note that the downdraft of the helicopter reduces the effects of air resistance on the falling life preserver, so that an acceleration equal to that of gravity is reasonable.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1788413\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1757046\">\n<p id=\"import-auto-id3572487\">A dolphin in an aquatic show jumps straight up out of the water at a velocity of 13.0 m\/s. (a) List the knowns in this problem. (b) How high does his body rise above the water? To solve this part, first note that the final velocity is now a known and identify its value. Then identify the unknown, and discuss how you chose the appropriate equation to solve for it. After choosing the equation, show your steps in solving for the unknown, checking units, and discuss whether the answer is reasonable. (c) How long is the dolphin in the air? Neglect any effects due to his size or orientation.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1722491\">\n<p id=\"import-auto-id2589095\">(a) [latex]a=-9\\text{.}{\\text{80 m\/s}}^{2}[\/latex]; [latex]{v}_{0}=\\text{13}\\text{.}\\text{0 m\/s}[\/latex]; [latex]{y}_{0}=\\text{0 m}[\/latex]<\/p>\n<p id=\"import-auto-id2589119\">(b) [latex]v=0\\text{m\/s}[\/latex]. Unknown is distance [latex]y[\/latex] to top of trajectory, where velocity is zero. Use equation [latex]{v}^{2}={v}_{0}^{2}+2a\\left(y-{y}_{0}\\right)[\/latex] because it contains all known values except for [latex]y[\/latex], so we can solve for [latex]y[\/latex]. Solving for [latex]y[\/latex] gives<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-id2418613\">[latex]\\begin{array}{lll}{v}^{2}-{v}_{0}^{2}&amp; =&amp; 2a\\left(y-{y}_{0}\\right)\\\\ \\frac{{v}^{2}-{v}_{0}^{2}}{2a}&amp; =&amp; y-{y}_{0}\\\\ y&amp; =&amp; {y}_{0}+\\frac{{v}^{2}-{v}_{0}^{2}}{2a}=0 m+\\frac{{\\left(\\text{0 m\/s}\\right)}^{2}-{\\left(\\text{13.0 m\/s}\\right)}^{2}}{2\\left(-\\text{9.80 m}{\\text{\/s}}^{2}\\right)}=\\text{8.62 m}\\end{array}[\/latex]<\/div>\n<p id=\"import-auto-id3567611\">Dolphins measure about 2 meters long and can jump several times their length out of the water, so this is a reasonable result.<\/p>\n<p id=\"import-auto-id3567615\">(c) [latex]2\\text{.}\\text{65 s}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1818111\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1758954\">\n<p id=\"import-auto-id3567649\">A swimmer bounces straight up from a diving board and falls feet first into a pool. She starts with a velocity of 4.00 m\/s, and her takeoff point is 1.80 m above the pool. (a) How long are her feet in the air? (b) What is her highest point above the board? (c) What is her velocity when her feet hit the water?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id4035152\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1746294\">\n<p id=\"import-auto-id1658343\">(a) Calculate the height of a cliff if it takes 2.35 s for a rock to hit the ground when it is thrown straight up from the cliff with an initial velocity of 8.00 m\/s. (b) How long would it take to reach the ground if it is thrown straight down with the same speed?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2367879\">\n<div class=\"bc-figure figure\" id=\"import-auto-id1658354\"><span data-type=\"media\" id=\"import-auto-id1658355\" data-alt=\"Path of a rock being thrown off of cliff. The rock moves up from the cliff top, reaches a transition point, and then falls down to the ground.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_02_07_05.jpg\" data-media-type=\"image\/jpg\" alt=\"Path of a rock being thrown off of cliff. The rock moves up from the cliff top, reaches a transition point, and then falls down to the ground.\" width=\"175\"><\/span><\/div>\n<p id=\"import-auto-id1658374\">(a) 8.26 m<\/p>\n<p id=\"import-auto-id1658376\">(b) 0.717 s<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2576295\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1742640\">\n<p id=\"import-auto-id1658283\">A very strong, but inept, shot putter puts the shot straight up vertically with an initial velocity of 11.0 m\/s. How long does he have to get out of the way if the shot was released at a height of 2.20 m, and he is 1.80 m tall?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3543404\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1471620\">\n<p id=\"import-auto-id1658299\">You throw a ball straight up with an initial velocity of 15.0 m\/s. It passes a tree branch on the way up at a height of 7.00 m. How much additional time will pass before the ball passes the tree branch on the way back down?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2042262\">\n<p id=\"import-auto-id1658312\">1.91 s<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id4076783\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1544876\">\n<p id=\"import-auto-id1658327\">A kangaroo can jump over an object 2.50 m high. (a) Calculate its vertical speed when it leaves the ground. (b) How long is it in the air? <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id4073110\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2360908\">\n<p id=\"import-auto-id4110058\">Standing at the base of one of the cliffs of Mt. Arapiles in Victoria, Australia, a hiker hears a rock break loose from a height of 105 m. He can\u2019t see the rock right away but then does, 1.50 s later. (a) How far above the hiker is the rock when he can see it? (b) How much time does he have to move before the rock hits his head?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id4065048\">\n<p id=\"import-auto-id4110070\">(a) 94.0 m <\/p>\n<p id=\"import-auto-id4110075\">(b) 3.13 s<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id776278\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3510883\">\n<p id=\"import-auto-id4110092\">An object is dropped from a height of 75.0 m above ground level. (a) Determine the distance traveled during the first second. (b) Determine the final velocity at which the object hits the ground. (c) Determine the distance traveled during the last second of motion before hitting the ground.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1798285\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1757237\">\n<p id=\"import-auto-id1893264\">There is a 250-m-high cliff at Half Dome in Yosemite National Park in California. Suppose a boulder breaks loose from the top of this cliff. (a) How fast will it be going when it strikes the ground? (b) Assuming a reaction time of 0.300 s, how long will a tourist at the bottom have to get out of the way after hearing the sound of the rock breaking loose (neglecting the height of the tourist, which would become negligible anyway if hit)? The speed of sound is 335 m\/s on this day.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1465838\">\n<p id=\"import-auto-id1893277\">(a) -70.0 m\/s (downward)<\/p>\n<p id=\"import-auto-id1893279\">(b) 6.10 s<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id4044798\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1758045\">\n<p id=\"import-auto-id1893300\">A ball is thrown straight up. It passes a 2.00-m-high window 7.50 m off the ground on its path up and takes 0.312 s to go past the window. What was the ball\u2019s initial velocity? Hint: First consider only the distance along the window, and solve for the ball's velocity at the bottom of the window. Next, consider only the distance from the ground to the bottom of the window, and solve for the initial velocity using the velocity at the bottom of the window as the final velocity.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id4048528\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id4122121\">\n<p id=\"import-auto-id1893315\">Suppose you drop a rock into a dark well and, using precision equipment, you measure the time for the sound of a splash to return. (a) Neglecting the time required for sound to travel up the well, calculate the distance to the water if the sound returns in 2.0000 s. (b) Now calculate the distance taking into account the time for sound to travel up the well. The speed of sound is 332.00 m\/s in this well.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1772027\">\n<p id=\"import-auto-id1724786\">(a) [latex]\\text{19}\\text{.}\\text{6 m}[\/latex]<\/p>\n<p id=\"import-auto-id1724802\">(b) [latex]\\text{18}\\text{.}\\text{5 m}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2561073\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id4020060\">\n<p id=\"import-auto-id2282109\">A steel ball is dropped onto a hard floor from a height of 1.50 m and rebounds to a height of 1.45 m. (a) Calculate its velocity just before it strikes the floor. (b) Calculate its velocity just after it leaves the floor on its way back up. (c) Calculate its acceleration during contact with the floor if that contact lasts 0.0800 ms [latex]\\left(8\\text{.}\\text{00}\u00d7{\\text{10}}^{-5}\\phantom{\\rule{0.25em}{0ex}}\\text{s}\\right)[\/latex]. (d) How much did the ball compress during its collision with the floor, assuming the floor is absolutely rigid?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2227967\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1934778\">\n<p id=\"import-auto-id2589141\">A coin is dropped from a hot-air balloon that is 300 m above the ground and rising at 10.0 m\/s upward. For the coin, find (a) the maximum height reached, (b) its position and velocity 4.00 s after being released, and (c) the time before it hits the ground.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id3557416\">\n<p id=\"import-auto-id2282130\">(a) 305 m<\/p>\n<p id=\"import-auto-id2282135\">(b) 262 m, -29.2 m\/s <\/p>\n<p id=\"import-auto-id2282137\">(c) 8.91 s<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3597625\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3502580\">\n<p id=\"import-auto-id1590185\">A soft tennis ball is dropped onto a hard floor from a height of 1.50 m and rebounds to a height of 1.10 m. (a) Calculate its velocity just before it strikes the floor. (b) Calculate its velocity just after it leaves the floor on its way back up. (c) Calculate its acceleration during contact with the floor if that contact lasts 3.50 ms [latex]\\left(3\\text{.}\\text{50}\u00d7{\\text{10}}^{-3}\\phantom{\\rule{0.25em}{0ex}}\\text{s}\\right)[\/latex]. (d) How much did the ball compress during its collision with the floor, assuming the floor is absolutely rigid?<\/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-id2801708\">\n<dt>free-fall<\/dt>\n<dd id=\"fs-id4121187\">the state of movement that results from gravitational force only<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id2801710\">\n<dt>acceleration due to gravity<\/dt>\n<dd id=\"fs-id2255729\">acceleration of an object as a result of gravity <\/dd>\n<\/dl>\n<\/div>\n\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<ul>\n<li>Describe the effects of gravity on objects in motion.<\/li>\n<li>Describe the motion of objects that are in free fall.<\/li>\n<li>Calculate the position and velocity of objects in free fall.<\/li>\n<\/ul>\n<\/div>\n<p id=\"import-auto-id2295392\">Falling objects form an interesting class of motion problems. For example, we can estimate the depth of a vertical mine shaft by dropping a rock into it and listening for the rock to hit the bottom. By applying the kinematics developed so far to falling objects, we can examine some interesting situations and learn much about gravity in the process.<\/p>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id4178141\">\n<h1 data-type=\"title\">Gravity<\/h1>\n<p id=\"import-auto-id2006874\">The most remarkable and unexpected fact about falling objects is that, if air resistance and friction are negligible, then in a given location all objects fall toward the center of Earth with the <em data-effect=\"italics\">same constant acceleration<\/em>, <em data-effect=\"italics\">independent of their mass<\/em>. This experimentally determined fact is unexpected, because we are so accustomed to the effects of air resistance and friction that we expect light objects to fall slower than heavy ones. <\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id4126662\">\n<div class=\"bc-figcaption figcaption\">A hammer and a feather will fall with the same constant acceleration if air resistance is considered negligible. This is a general characteristic of gravity not unique to Earth, as astronaut David R. Scott demonstrated on the Moon in 1971, where the acceleration due to gravity is only <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-deef0687018cf29e5c0df4de5e9dc581_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#55;&#32;&#109;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"74\" style=\"vertical-align: -4px;\" \/>.\n      <\/div>\n<p><span data-type=\"media\" id=\"import-auto-id2553814\" data-alt=\"Positions of a feather and hammer over time as they fall on the Moon. The feather and hammer are at the exact same position at each moment in time.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_02_07_00a.jpg\" data-media-type=\"image\/jpg\" alt=\"Positions of a feather and hammer over time as they fall on the Moon. The feather and hammer are at the exact same position at each moment in time.\" width=\"300\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id4066514\">In the real world, air resistance can cause a lighter object to fall slower than a heavier object of the same size. A tennis ball will reach the ground after a hard baseball dropped at the same time. (It might be difficult to observe the difference if the height is not large.) Air resistance opposes the motion of an object through the air, while friction between objects\u2014such as between clothes and a laundry chute or between a stone and a pool into which it is dropped\u2014also opposes motion between them. For the ideal situations of these first few chapters, an object <em data-effect=\"italics\">falling without air resistance or friction<\/em> is defined to be in <span data-type=\"term\" id=\"import-auto-id1714641\">free-fall<\/span>.<\/p>\n<p id=\"import-auto-id4141851\">The force of gravity causes objects to fall toward the center of Earth. The acceleration of free-falling objects is therefore called the <span data-type=\"term\" id=\"import-auto-id1707599\">acceleration due to gravity<\/span>. The acceleration due to gravity is <em data-effect=\"italics\">constant<\/em>, which means we can apply the kinematics equations to any falling object where air resistance and friction are negligible. This opens a broad class of interesting situations to us. The acceleration due to gravity is so important that its magnitude is given its own symbol, <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;\" \/>. It is constant at any given location on Earth and has the average value<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-636\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9dce5324fe9606bf0ac48884e371db52_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#61;&#57;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#56;&#48;&#32;&#109;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"112\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"import-auto-id3504907\">Although <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;\" \/> varies from <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-e81e146e372c2b95000e87691cf86784_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#55;&#56;&#32;&#109;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"75\" style=\"vertical-align: -4px;\" \/> to [latex]9\\text{.}{\\text{83 m\/s}}^{2}[\/latex], depending on latitude, altitude, underlying geological formations, and local topography, the average value of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-438c95347c421cf11dd029d83895c960_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#56;&#48;&#32;&#109;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"75\" style=\"vertical-align: -4px;\" \/> will be used in this text unless otherwise specified. The direction of the acceleration due to gravity is <em data-effect=\"italics\">downward (towards the center of Earth)<\/em>. In fact, its direction <em data-effect=\"italics\">defines<\/em> what we call vertical. Note that whether the acceleration <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;\" \/> in the kinematic equations has the value <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d47489b58944b0002ababe682569f735_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#43;&#103;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"23\" style=\"vertical-align: -4px;\" \/> or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-921df4d38393c35aba817c0206386e00_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#103;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: -4px;\" \/> depends on how we define our coordinate system. If we define the upward direction as positive, then <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ef9681d54515ea9b8ec82d7853f54cf6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#45;&#103;&#61;&#45;&#57;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#56;&#48;&#32;&#109;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"168\" style=\"vertical-align: -4px;\" \/>, and if we define the downward direction as positive, then <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a425364a01ab562e76e5da8f6365173c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#103;&#61;&#57;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#56;&#48;&#32;&#109;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"141\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id2854808\">\n<h1 data-type=\"title\">One-Dimensional Motion Involving Gravity<\/h1>\n<p id=\"import-auto-id1354046\">The best way to see the basic features of motion involving gravity is to start with the simplest situations and then progress toward more complex ones. So we start by considering straight up and down motion with no air resistance or friction. These assumptions mean that the velocity (if there is any) is vertical. If the object is dropped, we know the initial velocity is zero. Once the object has left contact with whatever held or threw it, the object is in free-fall. Under these circumstances, the motion is one-dimensional and has constant acceleration of magnitude <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;\" \/>. We will also represent vertical displacement with the symbol <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 use <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;\" \/> for horizontal displacement.<\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id1931991\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Kinematic Equations for Objects in Free-Fall where Acceleration = &#8211;<em data-effect=\"italics\">g<\/em><\/div>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2222965\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-bc8904023b7865f552b7c122b32f465e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;&#61;&#123;&#118;&#125;&#95;&#123;&#48;&#125;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#103;&#116;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"86\" style=\"vertical-align: -3px;\" \/><\/div>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id4048491\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-23e8ac9d44da7ed5b73a6eb09b6cc03a_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;&#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=\"149\" style=\"vertical-align: -6px;\" \/><\/div>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id4019890\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-42dce989624bc2d8de121f5c91e50cdc_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;&#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;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"160\" style=\"vertical-align: -5px;\" \/><\/div>\n<\/div>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id4067058\">\n<div data-type=\"title\" class=\"title\">Calculating Position and Velocity of a Falling Object: A Rock Thrown Upward<\/div>\n<p id=\"import-auto-id4149723\">A person standing on the edge of a high cliff throws a rock straight up with an initial velocity of 13.0 m\/s<em data-effect=\"italics\">.<\/em> The rock misses the edge of the cliff as it falls back to earth. Calculate the position and velocity of the rock 1.00 s, 2.00 s, and 3.00 s after it is thrown, neglecting the effects of air resistance.<\/p>\n<p id=\"import-auto-id4127381\"><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id1451572\">Draw a sketch.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id2188586\"><span data-type=\"media\" id=\"import-auto-id3561557\" data-alt=\"Velocity vector arrow pointing up in the positive y direction, labeled v sub 0 equals thirteen point 0 meters per second. Acceleration vector arrow pointing down in the negative y direction, labeled a equals negative 9 point 8 meters per second squared.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_02_07_00b.jpg\" data-media-type=\"image\/jpg\" alt=\"Velocity vector arrow pointing up in the positive y direction, labeled v sub 0 equals thirteen point 0 meters per second. Acceleration vector arrow pointing down in the negative y direction, labeled a equals negative 9 point 8 meters per second squared.\" width=\"350\" \/><\/span><\/div>\n<p id=\"import-auto-id1345513\">We are asked to determine the 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;\" \/> at various times. It is reasonable to 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. This problem involves one-dimensional motion in the vertical direction. We use plus and minus signs to indicate direction, with up being positive and down negative. Since up is positive, and the rock is thrown upward, the initial velocity must be positive too. The acceleration due to gravity is downward, so <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;\" \/> is negative. It is crucial that the initial velocity and the acceleration due to gravity have opposite signs. Opposite signs indicate that the acceleration due to gravity opposes the initial motion and will slow and eventually reverse it. <\/p>\n<p id=\"import-auto-id4095755\">Since we are asked for values of position and velocity at three times, we will refer to these as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-fcae814fd0a6c126f8e845b5cb49eb84_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#95;&#123;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"15\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2ae0996f672b3c41b449ed8af9d729b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"15\" style=\"vertical-align: -4px;\" \/>; <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6e09b16903e98422ba9e190869e1901c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#95;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: -4px;\" \/><\/em> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a846206709fd15e5d155a8daa46ab489_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#50;&#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-c62b3b881b951e32a285bb9a7e08691b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#95;&#123;&#51;&#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-e8744fef1cfe8f3c2e24d85460fe7ebc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"16\" style=\"vertical-align: -3px;\" \/>.<\/p>\n<p id=\"import-auto-id2217851\"><strong>Solution for Position <\/strong><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-fcae814fd0a6c126f8e845b5cb49eb84_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#95;&#123;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"15\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"import-auto-id1659278\">1. Identify the knowns. We know that <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;\" \/>; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-26766df785f5f0b2e0cb937da38c5a27_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#48;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#51;&#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=\"108\" style=\"vertical-align: -4px;\" \/>; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ef9681d54515ea9b8ec82d7853f54cf6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#45;&#103;&#61;&#45;&#57;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#56;&#48;&#32;&#109;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"168\" style=\"vertical-align: -4px;\" \/>; and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-dfb4b5f87f59403548b7874d4e144bcc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#61;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#32;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"74\" style=\"vertical-align: -1px;\" \/>. <\/p>\n<p id=\"import-auto-id3627669\">2. Identify the best equation to use. We will use <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a638447c2e3d3d389be04e2081adad36_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;&#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=\"149\" style=\"vertical-align: -6px;\" \/> because it includes only one unknown, <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;\" \/> (or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-fcae814fd0a6c126f8e845b5cb49eb84_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#95;&#123;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"15\" style=\"vertical-align: -4px;\" \/>, here),<em data-effect=\"italics\"> which is the value we want to find.<\/em><\/p>\n<p id=\"import-auto-id3602200\">3. Plug in the known values and solve for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-fcae814fd0a6c126f8e845b5cb49eb84_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#95;&#123;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"15\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2302522\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-aa846bfa3fddea3b6573a537dbbf96b2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#123;&#125;&#95;&#123;&#49;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#125;&#61;&#48;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#51;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#32;&#109;&#47;&#115;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#32;&#115;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#57;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#56;&#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;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#32;&#115;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#56;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#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;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"502\" style=\"vertical-align: -12px;\" \/><\/div>\n<p id=\"import-auto-id1772102\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id2111072\">The rock is 8.10 m above its starting point at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-247940e0f57fae06d71c8ebfdf83ef16_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#61;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"62\" style=\"vertical-align: -1px;\" \/> s, since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-8ab2312b1d2035be7752449268406bdb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#95;&#123;&#49;&#125;&#62;&#123;&#121;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"56\" style=\"vertical-align: -4px;\" \/>. It could be <em data-effect=\"italics\">moving<\/em> up or down; the only way to tell is to calculate <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2ae0996f672b3c41b449ed8af9d729b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"15\" style=\"vertical-align: -4px;\" \/> and find out if it is positive or negative. <\/p>\n<p id=\"import-auto-id2223740\"><strong>Solution for Velocity <\/strong><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2ae0996f672b3c41b449ed8af9d729b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"15\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"import-auto-id4138785\">1. Identify the knowns. We know that <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;\" \/>; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-26766df785f5f0b2e0cb937da38c5a27_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#48;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#51;&#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=\"108\" style=\"vertical-align: -4px;\" \/>; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ef9681d54515ea9b8ec82d7853f54cf6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#45;&#103;&#61;&#45;&#57;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#56;&#48;&#32;&#109;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"168\" style=\"vertical-align: -4px;\" \/>; and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-dfb4b5f87f59403548b7874d4e144bcc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#61;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#32;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"74\" style=\"vertical-align: -1px;\" \/>. We also know from the solution above that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-fb584e01392ac5c03b3c6512fcf7f92c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#95;&#123;&#49;&#125;&#61;&#56;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#32;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"92\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<p id=\"import-auto-id4138788\">2. Identify the best equation to use. The most straightforward is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-bc8904023b7865f552b7c122b32f465e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;&#61;&#123;&#118;&#125;&#95;&#123;&#48;&#125;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#103;&#116;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"86\" style=\"vertical-align: -3px;\" \/> (from <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;\" \/>, where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f256e4c5a7c88852c42fe460dd6a42c9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#103;&#114;&#97;&#118;&#105;&#116;&#97;&#116;&#105;&#111;&#110;&#97;&#108;&#32;&#97;&#99;&#99;&#101;&#108;&#101;&#114;&#97;&#116;&#105;&#111;&#110;&#125;&#61;&#45;&#103;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"276\" style=\"vertical-align: -4px;\" \/>).<\/p>\n<p id=\"import-auto-id1688801\">3. Plug in the knowns and solve. <\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id1688803\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6822a62cf6c87d25d022f7f2ce3503a8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#49;&#125;&#61;&#123;&#118;&#125;&#95;&#123;&#48;&#125;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#103;&#116;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#51;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#32;&#109;&#47;&#115;&#125;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#56;&#48;&#32;&#109;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#32;&#115;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#51;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#48;&#32;&#109;&#47;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"457\" style=\"vertical-align: -12px;\" \/><\/div>\n<p id=\"import-auto-id2025041\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id3524154\">The positive value for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2ae0996f672b3c41b449ed8af9d729b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"15\" style=\"vertical-align: -4px;\" \/> means that the rock is still heading upward at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0d6cdef670b092a5a4a7c77353d5bc47_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#61;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#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;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"73\" style=\"vertical-align: -1px;\" \/>. However, it has slowed from its original 13.0<em data-effect=\"italics\"> m\/s, as expected.<\/em><\/p>\n<p id=\"import-auto-id2034327\"><strong>Solution for Remaining Times<\/strong><\/p>\n<p id=\"import-auto-id2001485\">The procedures for calculating the position and velocity at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b855b104aff1fc45b5d2ceca4b0ca1ff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#61;&#50;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#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;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"73\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-42ba67598f6b8e1c1253180d9ee9c28d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#32;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"44\" style=\"vertical-align: 0px;\" \/> are the same as those above. The results are summarized in <a href=\"#eip-304\" class=\"autogenerated-content\">(Figure)<\/a> and illustrated in <a href=\"#import-auto-id4064055\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<table id=\"eip-304\" summary=\"Table with four columns showing the time, position, velocity, and acceleration of a rock thrown in the air. Times are listed in column one, and corresponding position, velocity, and acceleration are listed in the next three columns.\">\n<caption><span data-type=\"title\">Results<\/span><\/caption>\n<thead>\n<tr>\n<th data-align=\"center\">\n            Time, <em data-effect=\"italics\">t<\/em>\n            <\/th>\n<th data-align=\"center\">\n              Position, <em data-effect=\"italics\">y<\/em>\n            <\/th>\n<th data-align=\"center\">\n              Velocity, <em data-effect=\"italics\">v<\/em>\n            <\/th>\n<th data-align=\"center\">\n              Acceleration, <em data-effect=\"italics\">a<\/em>\n            <\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-8bb5c18b7b52508bdcf60f893cd7cb22_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#32;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"43\" style=\"vertical-align: -1px;\" \/><\/td>\n<td data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-95478552b7bbd444da67840e1a194516_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;&#49;&#48;&#32;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"52\" style=\"vertical-align: -1px;\" \/><\/td>\n<td data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-026960f6310ef948db3e809a6969b947_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#48;&#32;&#109;&#47;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"68\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a194c89aad483d940695df2e0b502ea7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#57;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#56;&#48;&#32;&#109;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"88\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f7d2ec3e6aa297318bbe7ba93e98bbd4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#32;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"44\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a2270781586910ffd8a4129f9d7018d9_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;&#52;&#48;&#32;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"52\" style=\"vertical-align: -1px;\" \/><\/td>\n<td data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-223f107b9f588e2cb505689d49ba09ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#54;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#48;&#32;&#109;&#47;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"81\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a194c89aad483d940695df2e0b502ea7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#57;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#56;&#48;&#32;&#109;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"88\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-42ba67598f6b8e1c1253180d9ee9c28d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#32;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"44\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-92419c0e10c2d74dd613d1dfbd8c3df6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#32;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"65\" style=\"vertical-align: -1px;\" \/><\/td>\n<td data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-442c185383a34c354b66c3f03737dd0b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#54;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#32;&#109;&#47;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"81\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a194c89aad483d940695df2e0b502ea7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#57;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#56;&#48;&#32;&#109;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"88\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"import-auto-id3537673\">Graphing the data helps us understand it more clearly.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id4064055\">\n<div class=\"bc-figcaption figcaption\">Vertical position, vertical velocity, and vertical acceleration vs. time for a rock thrown vertically up at the edge of a cliff. Notice that velocity changes linearly with time and that acceleration is constant. <em data-effect=\"italics\">Misconception Alert!<\/em> Notice that the position vs. time graph shows vertical position only. It is easy to get the impression that the graph shows some horizontal motion\u2014the shape of the graph looks like the path of a projectile. But this is not the case; the horizontal axis is <em data-effect=\"italics\">time<\/em>, not space. The actual path of the rock in space is straight up, and straight down.\n      <\/div>\n<p><span data-type=\"media\" id=\"import-auto-id4064056\" data-alt=\"Three panels showing three graphs. The top panel shows a graph of vertical position in meters versus time in seconds. The line begins at the origin and has a positive slope that decreases over time until it hits a turning point between seconds 1 and 2. After that it has a negative slope that increases over time. The middle panel shows a graph of velocity in meters per second versus time in seconds. The line is straight, with a negative slope, beginning at time zero velocity of thirteen meters per second and ending at time 3 seconds with a velocity just over negative sixteen meters per second. The bottom panel shows a graph of acceleration in meters per second squared versus time in seconds. The line is straight and flat at a y value of negative 9 point 80 meters per second squared from time 0 to time 3 seconds.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_02_06_01.jpg\" data-media-type=\"image\/jpg\" alt=\"Three panels showing three graphs. The top panel shows a graph of vertical position in meters versus time in seconds. The line begins at the origin and has a positive slope that decreases over time until it hits a turning point between seconds 1 and 2. After that it has a negative slope that increases over time. The middle panel shows a graph of velocity in meters per second versus time in seconds. The line is straight, with a negative slope, beginning at time zero velocity of thirteen meters per second and ending at time 3 seconds with a velocity just over negative sixteen meters per second. The bottom panel shows a graph of acceleration in meters per second squared versus time in seconds. The line is straight and flat at a y value of negative 9 point 80 meters per second squared from time 0 to time 3 seconds.\" width=\"300\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id1681385\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id2220081\">The interpretation of these results is important. At 1.00 s the rock is above its starting point and heading upward, since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-fcae814fd0a6c126f8e845b5cb49eb84_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#95;&#123;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"15\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2ae0996f672b3c41b449ed8af9d729b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"15\" style=\"vertical-align: -4px;\" \/> are both positive. At 2.00 s, the rock is still above its starting point, but the negative velocity means it is moving downward. At 3.00 s, both <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c62b3b881b951e32a285bb9a7e08691b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#95;&#123;&#51;&#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-e8744fef1cfe8f3c2e24d85460fe7ebc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"16\" style=\"vertical-align: -3px;\" \/> are negative, meaning the rock is below its starting point and continuing to move downward. Notice that when the rock is at its highest point (at 1.5 s), its velocity is zero, but its acceleration is still <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a194c89aad483d940695df2e0b502ea7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#57;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#56;&#48;&#32;&#109;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"88\" style=\"vertical-align: -4px;\" \/>. Its acceleration is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a194c89aad483d940695df2e0b502ea7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#57;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#56;&#48;&#32;&#109;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"88\" style=\"vertical-align: -4px;\" \/><sup> for the whole trip\u2014while it is moving up and while it is moving down. Note that the values 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;\" \/><em data-effect=\"italics\"> are the positions (or displacements) of the rock, not the total distances traveled. Finally, note that free-fall applies to upward motion as well as downward. Both have the same acceleration\u2014the acceleration due to gravity, which remains constant the entire time. Astronauts training in the famous Vomit Comet, for example, experience free-fall while arcing up as well as down, as we will discuss in more detail later.<\/em><\/sup><\/p>\n<\/div>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id1773192\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Making Connections: Take-Home Experiment\u2014Reaction Time<\/div>\n<p id=\"import-auto-id2293959\">A simple experiment can be done to determine your reaction time. Have a friend hold a ruler between your thumb and index finger, separated by about 1 cm. Note the mark on the ruler that is right between your fingers. Have your friend drop the ruler unexpectedly, and try to catch it between your two fingers. Note the new reading on the ruler. Assuming acceleration is that due to gravity, calculate your reaction time. How far would you travel in a car (moving at 30 m\/s) if the time it took your foot to go from the gas pedal to the brake was twice this reaction time?<\/p>\n<\/div>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id2186600\">\n<div data-type=\"title\" class=\"title\">Calculating Velocity of a Falling Object: A Rock Thrown Down<\/div>\n<p id=\"import-auto-id4097406\">What happens if the person on the cliff throws the rock straight down, instead of straight up? To explore this question, calculate the velocity of the rock when it is 5.10 m below the starting point, and has been thrown downward with an initial speed of 13.0 m\/s.<\/p>\n<p id=\"import-auto-id4097414\"><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id2150745\">Draw a sketch. <\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id2150750\"><span data-type=\"media\" id=\"import-auto-id2150751\" data-alt=\"Velocity vector arrow pointing down in the negative y direction and labeled v sub zero equals negative thirteen point 0 meters per second. Acceleration vector arrow also pointing down in the negative y direction, labeled a equals negative 9 point 80 meters per second squared.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_02_06_01a.jpg\" data-media-type=\"image\/jpg\" alt=\"Velocity vector arrow pointing down in the negative y direction and labeled v sub zero equals negative thirteen point 0 meters per second. Acceleration vector arrow also pointing down in the negative y direction, labeled a equals negative 9 point 80 meters per second squared.\" width=\"350\" \/><\/span><\/div>\n<p id=\"import-auto-id3604721\">Since up is positive, the final position of the rock will be negative because it finishes below the starting point at <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;\" \/>. Similarly, the initial velocity is downward and therefore negative, as is the acceleration due to gravity. We expect the final velocity to be negative since the rock will continue to move downward.<\/p>\n<p id=\"import-auto-id4110111\"><strong>Solution<\/strong><\/p>\n<p id=\"import-auto-id1788409\">1. Identify the knowns. <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;\" \/>;<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0b148d0d9028b84b767b41ede6010084_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#95;&#123;&#49;&#125;&#61;&#45;&#53;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#32;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"106\" style=\"vertical-align: -4px;\" \/>;<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-60754018522deebc2b74e9d24da6c846_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#48;&#125;&#61;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#51;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#48;&#32;&#109;&#47;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"122\" style=\"vertical-align: -4px;\" \/>;<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-e497d25a437c06f60b03a8c1e2ec32de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#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;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"168\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<p id=\"import-auto-id1782245\">2. Choose the kinematic equation that makes it easiest to solve the problem. The equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5ffbb09ded70147bc97ec2e99d3fc9eb_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;&#121;&#45;&#123;&#121;&#125;&#95;&#123;&#48;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"161\" style=\"vertical-align: -5px;\" \/> works well because the only unknown in it is <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 will plug <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-fcae814fd0a6c126f8e845b5cb49eb84_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#95;&#123;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"15\" style=\"vertical-align: -4px;\" \/> in 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;\" \/>.)<\/p>\n<p id=\"import-auto-id2025015\">3. Enter the known values<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2025017\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c3969c3973aeccae6683e015667ed0ff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#94;&#123;&#50;&#125;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#51;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#32;&#109;&#47;&#115;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#57;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#56;&#48;&#32;&#109;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#32;&#109;&#125;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#32;&#109;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#54;&#56;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#54;&#32;&#109;&#125;&#125;&#94;&#123;&#50;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"551\" style=\"vertical-align: -12px;\" \/><\/div>\n<p id=\"import-auto-id1763708\">where we have retained extra significant figures because this is an intermediate result.<\/p>\n<p id=\"import-auto-id1763705\">Taking the square root, and noting that a square root can be positive or negative, gives<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id1763704\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5ff59d230c17fa375c0fcb2b15702f78_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;&#61;&plusmn;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#54;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#52;&#32;&#109;&#47;&#115;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"105\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"import-auto-id1797916\">The negative root is chosen to indicate that the rock is still heading down. Thus,<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2563747\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3aecbb04fcb51d405d0338b9068559e8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;&#61;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#54;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#52;&#32;&#109;&#47;&#115;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"119\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"import-auto-id2561834\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id2563753\">Note that <em data-effect=\"italics\">this is exactly the same velocity the rock had at this position when it was thrown straight upward with the same initial speed<\/em>. (See <a href=\"#fs-id4067058\" class=\"autogenerated-content\">(Figure)<\/a> and <a href=\"#import-auto-id4173440\" class=\"autogenerated-content\">(Figure)<\/a>(a).) This is not a coincidental result. Because we only consider the acceleration due to gravity in this problem, the <em data-effect=\"italics\">speed<\/em> of a falling object depends only on its initial speed and its vertical position relative to the starting point. For example, if the velocity of the rock is calculated at a height of 8.10 m above the starting point (using the method from <a href=\"#fs-id4067058\" class=\"autogenerated-content\">(Figure)<\/a>) when the initial velocity is 13.0 m\/s straight up, a result of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-eaf6d32f34f359b5dc7fd4e3be604852_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&plusmn;&#51;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#48;&#32;&#109;&#47;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"68\" style=\"vertical-align: -4px;\" \/> is obtained. Here both signs are meaningful; the positive value occurs when the rock is at 8.10 m and heading up, and the negative value occurs when the rock is at 8.10 m and heading back down. It has the same <em data-effect=\"italics\">speed<\/em> but the opposite direction. <\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id4173440\">\n<div class=\"bc-figcaption figcaption\">(a) A person throws a rock straight up, as explored in <a href=\"#fs-id4067058\" class=\"autogenerated-content\">(Figure)<\/a>. The arrows are velocity vectors at 0, 1.00, 2.00, and 3.00 s. (b) A person throws a rock straight down from a cliff with the same initial speed as before, as in <a href=\"#fs-id2186600\" class=\"autogenerated-content\">(Figure)<\/a>. Note that at the same distance below the point of release, the rock has the same velocity in both cases. <\/div>\n<p><span data-type=\"media\" id=\"import-auto-id4173441\" data-alt=\"Two figures are shown. At left, a man standing on the edge of a cliff throws a rock straight up with an initial speed of thirteen meters per second. At right, the man throws the rock straight down with a speed of thirteen meters per second. In both figures, a line indicates the rock\u2019s trajectory. When the rock is thrown straight up, it has a speed of minus sixteen point four meters per second at minus five point one zero meters below the point where the man released the rock. When the rock is thrown straight down, the velocity is the same at this position.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_02_06_00b.jpg\" data-media-type=\"image\/jpg\" alt=\"Two figures are shown. At left, a man standing on the edge of a cliff throws a rock straight up with an initial speed of thirteen meters per second. At right, the man throws the rock straight down with a speed of thirteen meters per second. In both figures, a line indicates the rock\u2019s trajectory. When the rock is thrown straight up, it has a speed of minus sixteen point four meters per second at minus five point one zero meters below the point where the man released the rock. When the rock is thrown straight down, the velocity is the same at this position.\" width=\"450\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id4180408\">Another way to look at it is this: In <a href=\"#fs-id4067058\" class=\"autogenerated-content\">(Figure)<\/a>, the rock is thrown up with an initial velocity of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b5b764fba77b43681e4f8feeabbc3900_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#51;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#48;&#32;&#109;&#47;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"67\" style=\"vertical-align: -4px;\" \/>. It rises and then falls back down. When its position is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5e8ef70615fdaee8588017ac1fdd2da0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"42\" style=\"vertical-align: -4px;\" \/> on its way back down, its velocity is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-17223d2a490924ef6350b1bc4812f974_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#51;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#48;&#32;&#109;&#47;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"81\" style=\"vertical-align: -4px;\" \/>. That is, it has the same speed on its way down as on its way up. We would then expect its velocity at a position of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-75e5c1ab36220a8ee96617ef0db83ad0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#53;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#32;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"99\" style=\"vertical-align: -4px;\" \/> to be the same whether we have thrown it upwards at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-003134a5002fd54145487022d3f860b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#43;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#51;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#48;&#32;&#109;&#47;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"82\" style=\"vertical-align: -4px;\" \/> or thrown it downwards at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-17223d2a490924ef6350b1bc4812f974_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#51;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#48;&#32;&#109;&#47;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"81\" style=\"vertical-align: -4px;\" \/>. The velocity of the rock on its way down from <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5e8ef70615fdaee8588017ac1fdd2da0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"42\" style=\"vertical-align: -4px;\" \/> is the same whether we have thrown it up or down to start with, as long as the speed with which it was initially thrown is the same. <\/p>\n<\/div>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id3526422\">\n<div data-type=\"title\" class=\"title\">Find <em data-effect=\"italics\">g<\/em> from Data on a Falling Object<\/div>\n<p id=\"import-auto-id2585093\">The acceleration due to gravity on Earth differs slightly from place to place, depending on topography (e.g., whether you are on a hill or in a valley) and subsurface geology (whether there is dense rock like iron ore as opposed to light rock like salt beneath you.) The precise acceleration due to gravity can be calculated from data taken in an introductory physics laboratory course. An object, usually a metal ball for which air resistance is negligible, is dropped and the time it takes to fall a known distance is measured. See, for example, <a href=\"#import-auto-id4097254\" class=\"autogenerated-content\">(Figure)<\/a>. Very precise results can be produced with this method if sufficient care is taken in measuring the distance fallen and the elapsed time. <\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id4097254\">\n<div class=\"bc-figcaption figcaption\">Positions and velocities of a metal ball released from rest when air resistance is negligible. Velocity is seen to increase linearly with time while displacement increases with time squared. Acceleration is a constant and is equal to gravitational acceleration.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id4097255\" data-alt=\"Figure has four panels. The first panel (on the top) is an illustration of a ball falling toward the ground at intervals of one tenth of a second. The space between the vertical position of the ball at one time step and the next increases with each time step. At time equals 0, position and velocity are also 0. At time equals 0 point 1 seconds, y position equals negative 0 point 049 meters and velocity is negative 0 point 98 meters per second. At 0 point 5 seconds, y position is negative 1 point 225 meters and velocity is negative 4 point 90 meters per second. The second panel (in the middle) is a line graph of position in meters versus time in seconds. Line begins at the origin and slopes down with increasingly negative slope. The third panel (bottom left) is a line graph of velocity in meters per second versus time in seconds. Line is straight, beginning at the origin and with a constant negative slope. The fourth panel (bottom right) is a line graph of acceleration in meters per second squared versus time in seconds. Line is flat, at a constant y value of negative 9 point 80 meters per second squared.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_02_06_02.jpg\" data-media-type=\"image\/jpg\" alt=\"Figure has four panels. The first panel (on the top) is an illustration of a ball falling toward the ground at intervals of one tenth of a second. The space between the vertical position of the ball at one time step and the next increases with each time step. At time equals 0, position and velocity are also 0. At time equals 0 point 1 seconds, y position equals negative 0 point 049 meters and velocity is negative 0 point 98 meters per second. At 0 point 5 seconds, y position is negative 1 point 225 meters and velocity is negative 4 point 90 meters per second. The second panel (in the middle) is a line graph of position in meters versus time in seconds. Line begins at the origin and slopes down with increasingly negative slope. The third panel (bottom left) is a line graph of velocity in meters per second versus time in seconds. Line is straight, beginning at the origin and with a constant negative slope. The fourth panel (bottom right) is a line graph of acceleration in meters per second squared versus time in seconds. Line is flat, at a constant y value of negative 9 point 80 meters per second squared.\" width=\"550\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id4146639\">Suppose the ball falls 1.0000 m in 0.45173 s. Assuming the ball is not affected by air resistance, what is the precise acceleration due to gravity at this location?<\/p>\n<p id=\"import-auto-id4146642\"><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id4051154\">Draw a sketch. <\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id4051158\"><span data-type=\"media\" id=\"import-auto-id4051159\" data-alt=\"The figure shows a green dot labeled v sub zero equals zero meters per second, a purple downward pointing arrow labeled a equals question mark, and an x y coordinate system with the y axis pointing vertically up and the x axis pointing horizontally to the right.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_02_06_02b.jpg\" data-media-type=\"image\/jpg\" alt=\"The figure shows a green dot labeled v sub zero equals zero meters per second, a purple downward pointing arrow labeled a equals question mark, and an x y coordinate system with the y axis pointing vertically up and the x axis pointing horizontally to the right.\" width=\"350\" \/><\/span><\/div>\n<p id=\"import-auto-id4025931\">We need to solve for acceleration <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;\" \/>. Note that in this case, displacement is downward and therefore negative, as is acceleration. <\/p>\n<p id=\"import-auto-id1551030\"><strong>Solution<\/strong><\/p>\n<p id=\"import-auto-id1551034\">1. Identify the knowns. <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;\" \/>;<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-820545f5d4e24a85704cc547942cc621_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#48;&#48;&#48;&#48;&#32;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"117\" style=\"vertical-align: -4px;\" \/>;<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9aa157f24bd559e663e387214196d66d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#61;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#52;&#53;&#49;&#55;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"89\" style=\"vertical-align: -1px;\" \/>; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-fe5a0f198037f16ce4688b674eeab213_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#48;&#125;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"49\" style=\"vertical-align: -3px;\" \/>. <\/p>\n<p id=\"import-auto-id4043850\">2. Choose the equation that allows you to solve for <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;\" \/> using the known values.<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id3578358\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a638447c2e3d3d389be04e2081adad36_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;&#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=\"149\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id4160974\">3. Substitute 0 for <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;\" \/> and rearrange the equation to solve for <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;\" \/>. Substituting 0 for <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;\" \/> yields<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id3538274\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-82f6310c301a555c4fafc7c2bf0253c2_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;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#116;&#125;&#125;&#94;&#123;&#50;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"109\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id3504490\">Solving for <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;\" \/> gives<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id3504484\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-cae391e59d639bcad462b26371e85d9e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#123;&#121;&#125;&#95;&#123;&#48;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#123;&#116;&#125;&#94;&#123;&#50;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"90\" style=\"vertical-align: -7px;\" \/><\/div>\n<p id=\"import-auto-id4128814\">4. Substitute known values yields<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2571438\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b625be33b65441647ce23b4aab4eee8a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#48;&#48;&#32;&#109;&#32;&#45;&#32;&#48;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#53;&#49;&#55;&#51;&#32;&#115;&#125;&#123;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#61;&#45;&#57;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#56;&#48;&#49;&#48;&#32;&#109;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"31\" width=\"283\" style=\"vertical-align: -9px;\" \/><\/div>\n<p id=\"import-auto-id4129554\">so, because <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-50b7b492310ce914505222ec73dfdab4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#45;&#103;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"56\" style=\"vertical-align: -4px;\" \/> with the directions we have chosen,<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id1706716\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2255580b55d3ac8c1cbfaa53e329d8a6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#61;&#57;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#56;&#48;&#49;&#48;&#32;&#109;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"130\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"import-auto-id2018012\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id2007329\">The negative value for <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;\" \/> indicates that the gravitational acceleration is downward, as expected. We expect the value to be somewhere around the average value of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-438c95347c421cf11dd029d83895c960_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#56;&#48;&#32;&#109;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"75\" style=\"vertical-align: -4px;\" \/>, so <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ed1e4f1de92049ebb5e1b33040ca76a8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#56;&#48;&#49;&#48;&#32;&#109;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"93\" style=\"vertical-align: -4px;\" \/> makes sense. Since the data going into the calculation are relatively precise, this value for <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;\" \/> is more precise than the average value of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-438c95347c421cf11dd029d83895c960_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#56;&#48;&#32;&#109;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"75\" style=\"vertical-align: -4px;\" \/>; it represents the local value for the acceleration due to gravity.<\/p>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id4172780\" data-element-type=\"check-understanding\" data-label=\"\">\n<div data-type=\"title\">Check Your Understanding<\/div>\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1747475\">\n<p id=\"import-auto-id2325444\">A chunk of ice breaks off a glacier and falls 30.0 meters before it hits the water. Assuming it falls freely (there is no air resistance), how long does it take to hit the water?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1652531\">\n<p id=\"import-auto-id2358780\">We know that initial position <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;\" \/>, final position <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-69817d0fcc23a5085818fbfc9d7b8532_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#51;&#48;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#32;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"85\" style=\"vertical-align: -4px;\" \/>, and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ef9681d54515ea9b8ec82d7853f54cf6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#45;&#103;&#61;&#45;&#57;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#56;&#48;&#32;&#109;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"168\" style=\"vertical-align: -4px;\" \/>. We can then use the equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a638447c2e3d3d389be04e2081adad36_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;&#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=\"149\" style=\"vertical-align: -6px;\" \/> to 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;\" \/>. Inserting <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-50b7b492310ce914505222ec73dfdab4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#45;&#103;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"56\" style=\"vertical-align: -4px;\" \/>, we obtain<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2358783\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6a69b885b98cf6595529946dcf63d691_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;&#121;&#38;&#32;&#61;&#38;&#32;&#48;&#43;&#48;&#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;&#92;&#32;&#123;&#116;&#125;&#94;&#123;&#50;&#125;&#38;&#32;&#61;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#121;&#125;&#123;&#45;&#103;&#125;&#92;&#92;&#32;&#116;&#38;&#32;&#61;&#38;&#32;&plusmn;&#92;&#115;&#113;&#114;&#116;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#121;&#125;&#123;&#45;&#103;&#125;&#125;&#61;&plusmn;&#92;&#115;&#113;&#114;&#116;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#48;&#46;&#48;&#32;&#109;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#45;&#57;&#46;&#56;&#48;&#32;&#109;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;&#125;&#125;&#61;&plusmn;&#92;&#115;&#113;&#114;&#116;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#46;&#49;&#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;&#123;&#115;&#125;&#94;&#123;&#50;&#125;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#46;&#52;&#55;&#32;&#115;&#125;&#92;&#97;&#112;&#112;&#114;&#111;&#120;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#46;&#53;&#32;&#115;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"81\" width=\"431\" style=\"vertical-align: -36px;\" \/><\/div>\n<p id=\"import-auto-id2307769\">where we take the positive value as the physically relevant answer. Thus, it takes about 2.5 seconds for the piece of ice to hit the water.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id2006949\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">PhET Explorations: Equation Grapher<\/div>\n<p id=\"import-auto-id2044902\">Learn about graphing polynomials. The shape of the curve changes as the constants are adjusted. View the curves for the individual terms (e.g. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5f2c6badfc838e57b5951af0dfa55b8c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#98;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"52\" style=\"vertical-align: -4px;\" \/>) to see how they add to generate the polynomial curve.<\/p>\n<div class=\"bc-figure figure\" id=\"eip-id2087188\">\n<div class=\"bc-figcaption figcaption\"><a href=\"\/resources\/e6ee9717e3c9be4b4c32e56fa95242ccc73d2262\/equation-grapher_en.jar\">Equation Grapher<\/a><\/div>\n<p><span data-type=\"media\" id=\"Phet_module_2.7\" data-alt=\"\"><a href=\"\/resources\/e6ee9717e3c9be4b4c32e56fa95242ccc73d2262\/equation-grapher_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>\n<div class=\"section-summary\" data-depth=\"1\" id=\"fs-id1822906\">\n<h1 data-type=\"title\">Section Summary<\/h1>\n<ul id=\"fs-id3611205\">\n<li id=\"import-auto-id1715211\">An object in free-fall experiences constant acceleration if air resistance is negligible.<\/li>\n<li id=\"import-auto-id1715213\">On Earth, all free-falling objects have an 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;\" \/>, which averages\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id3547826\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5d426af7166ed063a2ef03e897a1d4f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#61;&#57;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#56;&#48;&#32;&#109;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"112\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/li>\n<li id=\"import-auto-id2150922\">Whether the acceleration <em data-effect=\"italics\">a <\/em>should be taken as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d47489b58944b0002ababe682569f735_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#43;&#103;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"23\" style=\"vertical-align: -4px;\" \/> or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-921df4d38393c35aba817c0206386e00_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#103;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: -4px;\" \/> is determined by your choice of coordinate system. If you choose the upward direction as positive, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-e497d25a437c06f60b03a8c1e2ec32de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#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;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"168\" style=\"vertical-align: -4px;\" \/> is negative. In the opposite case, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-93bb09a8925d4909b07ca44886e9db94_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#43;&#103;&#125;&#61;&#57;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#56;&#48;&#32;&#109;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"155\" style=\"vertical-align: -4px;\" \/> is positive. Since acceleration is constant, the kinematic equations above can be applied with the appropriate <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d47489b58944b0002ababe682569f735_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#43;&#103;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"23\" style=\"vertical-align: -4px;\" \/> or<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-921df4d38393c35aba817c0206386e00_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#103;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: -4px;\" \/> substituted for <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;\" \/>.<\/li>\n<li id=\"import-auto-id4051701\">For objects in free-fall, up is normally taken as positive for displacement, velocity, and acceleration.<\/li>\n<\/ul>\n<\/div>\n<div class=\"conceptual-questions\" data-depth=\"1\" id=\"fs-id1358164\" data-element-type=\"conceptual-questions\">\n<h1 data-type=\"title\">Conceptual Questions<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1427339\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id4095246\">\n<p id=\"import-auto-id2271017\">What is the acceleration of a rock thrown straight upward on the way up? At the top of its flight? On the way down?  <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3606158\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1713184\">\n<p id=\"import-auto-id2271030\">An object that is thrown straight up falls back to Earth. This is one-dimensional motion. (a) When is its velocity zero? (b) Does its velocity change direction? (c) Does the acceleration due to gravity have the same sign on the way up as on the way down? <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2044867\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2006404\">\n<p id=\"import-auto-id2281895\">Suppose you throw a rock nearly straight up at a coconut in a palm tree, and the rock misses on the way up but hits the coconut on the way down. Neglecting air resistance, how does the speed of the rock when it hits the coconut on the way down compare with what it would have been if it had hit the coconut on the way up? Is it more likely to dislodge the coconut on the way up or down? Explain. <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1773324\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2300211\">\n<p id=\"import-auto-id3728320\">If an object is thrown straight up and air resistance is negligible, then its speed when it returns to the starting point is the same as when it was released. If air resistance were not negligible, how would its speed upon return compare with its initial speed? How would the maximum height to which it rises be affected? <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1776230\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2158441\">\n<p id=\"import-auto-id3639568\">The severity of a fall depends on your speed when you strike the ground. All factors but the acceleration due to gravity being the same, how many times higher could a safe fall on the Moon be than on Earth (gravitational acceleration on the Moon is about 1\/6 that of the Earth)? <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2271654\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id4018154\">\n<p id=\"import-auto-id3547965\">How many times higher could an astronaut jump on the Moon than on Earth if his takeoff speed is the same in both locations (gravitational acceleration on the Moon is about 1\/6 of <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;\" \/> on Earth)? <\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"problems-exercises\" data-depth=\"1\" id=\"fs-id3514521\" data-element-type=\"problems-exercises\">\n<h1 data-type=\"title\">Problems &amp; Exercises<\/h1>\n<p id=\"eip-150\">Assume air resistance is negligible unless otherwise stated.<\/p>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1516821\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2558696\">\n<p id=\"import-auto-id1658423\">Calculate the displacement and velocity at times of (a) 0.500, (b) 1.00, (c) 1.50, and (d) 2.00 s for a ball thrown straight up with an initial velocity of 15.0 m\/s. Take the point of release to be <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;\" \/>.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1722520\">\n<p id=\"import-auto-id2303239\">(a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-cb0f082410cae71e089a77e70313a065_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#95;&#123;&#49;&#125;&#61;&#54;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#56;&#32;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"92\" style=\"vertical-align: -4px;\" \/>; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4541cb0837cab03f5d1a43a5476c48da_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#49;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#32;&#109;&#47;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"108\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"import-auto-id1725030\">(b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-81b9a72f64794a99a04bc1726e8790f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#95;&#123;&#50;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#32;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"92\" style=\"vertical-align: -4px;\" \/>; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-485e5d8988b0d82ba3b3c0c2fadc3c49_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#50;&#125;&#61;&#53;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#48;&#32;&#109;&#47;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"108\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"import-auto-id1658704\">(c) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-230d557f6525b0a79414d6856f54123b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#95;&#123;&#51;&#125;&#61;&#49;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#53;&#32;&#109;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"87\" style=\"vertical-align: -4px;\" \/>; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3d271abf05fc940225a5ad9dbb318a47_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#51;&#125;&#61;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#51;&#48;&#48;&#32;&#109;&#47;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"117\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"import-auto-id1725057\">(d) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f934311c882b37fe38c4d9afbdd3d935_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#95;&#123;&#52;&#125;&#61;&#49;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#52;&#32;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"92\" style=\"vertical-align: -4px;\" \/>; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6bea208d776b7f47d90bf1a75c7bf939_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#52;&#125;&#61;&#45;&#52;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#54;&#48;&#32;&#109;&#47;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"122\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1746555\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id4057791\">\n<p id=\"import-auto-id3728159\">Calculate the displacement and velocity at times of (a) 0.500, (b) 1.00, (c) 1.50, (d) 2.00, and (e) 2.50 s for a rock thrown straight down with an initial velocity of 14.0 m\/s from the Verrazano Narrows Bridge in New York City. The roadway of this bridge is 70.0 m above the water.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1781525\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1781526\">\n<p id=\"import-auto-id2354922\">A basketball referee tosses the ball straight up for the starting tip-off. At what velocity must a basketball player leave the ground to rise 1.25 m above the floor in an attempt to get the ball?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2300377\">\n<p id=\"import-auto-id2354930\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-39d06fbe1a9eec86c58766f6181d0fa1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#48;&#125;&#61;&#52;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#53;&#32;&#109;&#47;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"108\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1582773\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2593341\">\n<p id=\"import-auto-id1590258\">A rescue helicopter is hovering over a person whose boat has sunk. One of the rescuers throws a life preserver straight down to the victim with an initial velocity of 1.40 m\/s and observes that it takes 1.8 s to reach the water. (a) List the knowns in this problem. (b) How high above the water was the preserver released? Note that the downdraft of the helicopter reduces the effects of air resistance on the falling life preserver, so that an acceleration equal to that of gravity is reasonable.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1788413\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1757046\">\n<p id=\"import-auto-id3572487\">A dolphin in an aquatic show jumps straight up out of the water at a velocity of 13.0 m\/s. (a) List the knowns in this problem. (b) How high does his body rise above the water? To solve this part, first note that the final velocity is now a known and identify its value. Then identify the unknown, and discuss how you chose the appropriate equation to solve for it. After choosing the equation, show your steps in solving for the unknown, checking units, and discuss whether the answer is reasonable. (c) How long is the dolphin in the air? Neglect any effects due to his size or orientation.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1722491\">\n<p id=\"import-auto-id2589095\">(a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6ec47781c743c8829d45d032d654ca91_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#45;&#57;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#56;&#48;&#32;&#109;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"122\" style=\"vertical-align: -4px;\" \/>; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-26766df785f5f0b2e0cb937da38c5a27_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#48;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#51;&#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=\"108\" style=\"vertical-align: -4px;\" \/>; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-21cf53158f6edd693650ae5d5c571e36_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#95;&#123;&#48;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#32;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"70\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"import-auto-id2589119\">(b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d75ce5623d1bd92df92c3772b5c890b5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;&#61;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#47;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"73\" style=\"vertical-align: -4px;\" \/>. Unknown is distance <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;\" \/> to top of trajectory, where velocity is zero. Use equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5ffbb09ded70147bc97ec2e99d3fc9eb_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;&#121;&#45;&#123;&#121;&#125;&#95;&#123;&#48;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"161\" style=\"vertical-align: -5px;\" \/> because it contains all known values except 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;\" \/>, so we can solve 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;\" \/>. 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-id2418613\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-41c14cddfb908d8a653c7c25f2da745a_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;&#123;&#118;&#125;&#94;&#123;&#50;&#125;&#45;&#123;&#118;&#125;&#95;&#123;&#48;&#125;&#94;&#123;&#50;&#125;&#38;&#32;&#61;&#38;&#32;&#50;&#97;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#123;&#121;&#125;&#95;&#123;&#48;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#92;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#118;&#125;&#94;&#123;&#50;&#125;&#45;&#123;&#118;&#125;&#95;&#123;&#48;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#50;&#97;&#125;&#38;&#32;&#61;&#38;&#32;&#121;&#45;&#123;&#121;&#125;&#95;&#123;&#48;&#125;&#92;&#92;&#32;&#121;&#38;&#32;&#61;&#38;&#32;&#123;&#121;&#125;&#95;&#123;&#48;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#118;&#125;&#94;&#123;&#50;&#125;&#45;&#123;&#118;&#125;&#95;&#123;&#48;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#50;&#97;&#125;&#61;&#48;&#32;&#109;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#32;&#109;&#47;&#115;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#45;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#51;&#46;&#48;&#32;&#109;&#47;&#115;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#46;&#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;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#56;&#46;&#54;&#50;&#32;&#109;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"83\" width=\"466\" style=\"vertical-align: -37px;\" \/><\/div>\n<p id=\"import-auto-id3567611\">Dolphins measure about 2 meters long and can jump several times their length out of the water, so this is a reasonable result.<\/p>\n<p id=\"import-auto-id3567615\">(c) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-af0257bace7c7fa293e9f47c195f929a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#53;&#32;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"44\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1818111\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1758954\">\n<p id=\"import-auto-id3567649\">A swimmer bounces straight up from a diving board and falls feet first into a pool. She starts with a velocity of 4.00 m\/s, and her takeoff point is 1.80 m above the pool. (a) How long are her feet in the air? (b) What is her highest point above the board? (c) What is her velocity when her feet hit the water?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id4035152\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1746294\">\n<p id=\"import-auto-id1658343\">(a) Calculate the height of a cliff if it takes 2.35 s for a rock to hit the ground when it is thrown straight up from the cliff with an initial velocity of 8.00 m\/s. (b) How long would it take to reach the ground if it is thrown straight down with the same speed?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2367879\">\n<div class=\"bc-figure figure\" id=\"import-auto-id1658354\"><span data-type=\"media\" id=\"import-auto-id1658355\" data-alt=\"Path of a rock being thrown off of cliff. The rock moves up from the cliff top, reaches a transition point, and then falls down to the ground.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_02_07_05.jpg\" data-media-type=\"image\/jpg\" alt=\"Path of a rock being thrown off of cliff. The rock moves up from the cliff top, reaches a transition point, and then falls down to the ground.\" width=\"175\" \/><\/span><\/div>\n<p id=\"import-auto-id1658374\">(a) 8.26 m<\/p>\n<p id=\"import-auto-id1658376\">(b) 0.717 s<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2576295\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1742640\">\n<p id=\"import-auto-id1658283\">A very strong, but inept, shot putter puts the shot straight up vertically with an initial velocity of 11.0 m\/s. How long does he have to get out of the way if the shot was released at a height of 2.20 m, and he is 1.80 m tall?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3543404\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1471620\">\n<p id=\"import-auto-id1658299\">You throw a ball straight up with an initial velocity of 15.0 m\/s. It passes a tree branch on the way up at a height of 7.00 m. How much additional time will pass before the ball passes the tree branch on the way back down?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2042262\">\n<p id=\"import-auto-id1658312\">1.91 s<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id4076783\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1544876\">\n<p id=\"import-auto-id1658327\">A kangaroo can jump over an object 2.50 m high. (a) Calculate its vertical speed when it leaves the ground. (b) How long is it in the air? <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id4073110\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2360908\">\n<p id=\"import-auto-id4110058\">Standing at the base of one of the cliffs of Mt. Arapiles in Victoria, Australia, a hiker hears a rock break loose from a height of 105 m. He can\u2019t see the rock right away but then does, 1.50 s later. (a) How far above the hiker is the rock when he can see it? (b) How much time does he have to move before the rock hits his head?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id4065048\">\n<p id=\"import-auto-id4110070\">(a) 94.0 m <\/p>\n<p id=\"import-auto-id4110075\">(b) 3.13 s<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id776278\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3510883\">\n<p id=\"import-auto-id4110092\">An object is dropped from a height of 75.0 m above ground level. (a) Determine the distance traveled during the first second. (b) Determine the final velocity at which the object hits the ground. (c) Determine the distance traveled during the last second of motion before hitting the ground.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1798285\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1757237\">\n<p id=\"import-auto-id1893264\">There is a 250-m-high cliff at Half Dome in Yosemite National Park in California. Suppose a boulder breaks loose from the top of this cliff. (a) How fast will it be going when it strikes the ground? (b) Assuming a reaction time of 0.300 s, how long will a tourist at the bottom have to get out of the way after hearing the sound of the rock breaking loose (neglecting the height of the tourist, which would become negligible anyway if hit)? The speed of sound is 335 m\/s on this day.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1465838\">\n<p id=\"import-auto-id1893277\">(a) -70.0 m\/s (downward)<\/p>\n<p id=\"import-auto-id1893279\">(b) 6.10 s<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id4044798\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1758045\">\n<p id=\"import-auto-id1893300\">A ball is thrown straight up. It passes a 2.00-m-high window 7.50 m off the ground on its path up and takes 0.312 s to go past the window. What was the ball\u2019s initial velocity? Hint: First consider only the distance along the window, and solve for the ball&#8217;s velocity at the bottom of the window. Next, consider only the distance from the ground to the bottom of the window, and solve for the initial velocity using the velocity at the bottom of the window as the final velocity.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id4048528\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id4122121\">\n<p id=\"import-auto-id1893315\">Suppose you drop a rock into a dark well and, using precision equipment, you measure the time for the sound of a splash to return. (a) Neglecting the time required for sound to travel up the well, calculate the distance to the water if the sound returns in 2.0000 s. (b) Now calculate the distance taking into account the time for sound to travel up the well. The speed of sound is 332.00 m\/s in this well.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1772027\">\n<p id=\"import-auto-id1724786\">(a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-184666161ff217e38a9dae8ef0ae3f71_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#57;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#32;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"51\" style=\"vertical-align: -1px;\" \/><\/p>\n<p id=\"import-auto-id1724802\">(b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1d3406c039cb0dfbcf9163d062cd80bd_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;&#53;&#32;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"51\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2561073\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id4020060\">\n<p id=\"import-auto-id2282109\">A steel ball is dropped onto a hard floor from a height of 1.50 m and rebounds to a height of 1.45 m. (a) Calculate its velocity just before it strikes the floor. (b) Calculate its velocity just after it leaves the floor on its way back up. (c) Calculate its acceleration during contact with the floor if that contact lasts 0.0800 ms <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-01d4a278d07be5e3a8fd68cfd0428275_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#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;&#45;&#53;&#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;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"91\" style=\"vertical-align: -7px;\" \/>. (d) How much did the ball compress during its collision with the floor, assuming the floor is absolutely rigid?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2227967\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1934778\">\n<p id=\"import-auto-id2589141\">A coin is dropped from a hot-air balloon that is 300 m above the ground and rising at 10.0 m\/s upward. For the coin, find (a) the maximum height reached, (b) its position and velocity 4.00 s after being released, and (c) the time before it hits the ground.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id3557416\">\n<p id=\"import-auto-id2282130\">(a) 305 m<\/p>\n<p id=\"import-auto-id2282135\">(b) 262 m, -29.2 m\/s <\/p>\n<p id=\"import-auto-id2282137\">(c) 8.91 s<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3597625\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3502580\">\n<p id=\"import-auto-id1590185\">A soft tennis ball is dropped onto a hard floor from a height of 1.50 m and rebounds to a height of 1.10 m. (a) Calculate its velocity just before it strikes the floor. (b) Calculate its velocity just after it leaves the floor on its way back up. (c) Calculate its acceleration during contact with the floor if that contact lasts 3.50 ms <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2e668ca1d6ecee31a29d07b4befd6a86_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#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;&#115;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"91\" style=\"vertical-align: -7px;\" \/>. (d) How much did the ball compress during its collision with the floor, assuming the floor is absolutely rigid?<\/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-id2801708\">\n<dt>free-fall<\/dt>\n<dd id=\"fs-id4121187\">the state of movement that results from gravitational force only<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id2801710\">\n<dt>acceleration due to gravity<\/dt>\n<dd id=\"fs-id2255729\">acceleration of an object as a result of gravity <\/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-118","chapter","type-chapter","status-publish","hentry","license-all-rights-reserved"],"part":58,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/118","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\/118\/revisions"}],"predecessor-version":[{"id":119,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/118\/revisions\/119"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/parts\/58"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/118\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/media?parent=118"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapter-type?post=118"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/contributor?post=118"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/license?post=118"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}