{"id":143,"date":"2017-10-27T16:29:00","date_gmt":"2017-10-27T16:29:00","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/chapter\/graphical-analysis-of-one-dimensional-motion\/"},"modified":"2017-11-08T03:23:53","modified_gmt":"2017-11-08T03:23:53","slug":"graphical-analysis-of-one-dimensional-motion","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/chapter\/graphical-analysis-of-one-dimensional-motion\/","title":{"raw":"Graphical Analysis of One-Dimensional Motion","rendered":"Graphical Analysis of One-Dimensional Motion"},"content":{"raw":"\n<div class=\"textbox learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<ul>\n<li>Describe a straight-line graph in terms of its slope and <span data-type=\"foreign\">y<\/span>-intercept.<\/li>\n<li>Determine average velocity or instantaneous velocity from a graph of position vs. time.<\/li>\n<li>Determine average or instantaneous acceleration from a graph of velocity vs. time.<\/li>\n<li>Derive a graph of velocity vs. time from a graph of position vs. time.<\/li>\n<li>Derive a graph of acceleration vs. time from a graph of velocity vs. time.<\/li>\n<\/ul>\n<\/div>\n<p id=\"import-auto-id1568723\">A graph, like a picture, is worth a thousand words. Graphs not only contain numerical information; they also reveal relationships between physical quantities. This section uses graphs of displacement, velocity, and acceleration versus time to illustrate one-dimensional kinematics.<\/p>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1396690\">\n<h1 data-type=\"title\">Slopes and General Relationships<\/h1>\n<p id=\"import-auto-id4131770\">First note that graphs in this text have perpendicular axes, one horizontal and the other vertical. When two physical quantities are plotted against one another in such a graph, the horizontal axis is usually considered to be an <span data-type=\"term\" id=\"import-auto-id1690042\">independent variable<\/span> and the vertical axis a <span data-type=\"term\" id=\"import-auto-id2013112\">dependent variable<\/span>. If we call the horizontal axis the [latex]x[\/latex]-axis and the vertical axis the [latex]y[\/latex]-axis, as in <a href=\"#import-auto-id2359358\" class=\"autogenerated-content\">(Figure)<\/a>, a straight-line graph has the general form<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id4175150\">[latex]y=\\text{mx}+b.[\/latex]<\/div>\n<p id=\"import-auto-id1433182\">Here [latex]m[\/latex] is the <span data-type=\"term\" id=\"import-auto-id1773074\">slope<\/span>, defined to be the rise divided by the run (as seen in the figure) of the straight line. The letter [latex]b[\/latex] is used for the <span data-type=\"term\" id=\"import-auto-id2955215\"><em data-effect=\"italics\">y<\/em>-intercept<\/span>, which is the point at which the line crosses the vertical axis.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id2359358\">\n<div class=\"bc-figcaption figcaption\">A straight-line graph. The equation for a straight line is [latex]y=\\text{mx}+b[\/latex]<br>\n    .<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1534567\" data-alt=\"Graph of a straight-line sloping up at about 40 degrees.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_02_07_01.jpg\" data-media-type=\"image\/jpg\" alt=\"Graph of a straight-line sloping up at about 40 degrees.\" width=\"300\"><\/span><\/p><\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id2201114\">\n<h1 data-type=\"title\">Graph of Displacement vs. Time (<em data-effect=\"italics\">a<\/em> = 0, so <em data-effect=\"italics\">v<\/em> is constant)<\/h1>\n<p id=\"import-auto-id1768816\">Time is usually an independent variable that other quantities, such as displacement, depend upon. A graph of displacement versus time would, thus, have [latex]x[\/latex] on the vertical axis and [latex]t[\/latex] on the horizontal axis.   <a href=\"#import-auto-id2574769\" class=\"autogenerated-content\">(Figure)<\/a> is just such a straight-line graph. It shows a graph of displacement versus time for a jet-powered car on a very flat dry lake bed in Nevada.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id2574769\">\n<div class=\"bc-figcaption figcaption\">Graph of displacement versus time for a jet-powered car on the Bonneville Salt Flats. <\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1599004\" data-alt=\"Line graph of jet car displacement in meters versus time in seconds. The line is straight with a positive slope. The y intercept is four hundred meters. The total change in time is eight point zero seconds. The initial position is four hundred meters. The final position is two thousand meters.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_02_07_02.jpg\" data-media-type=\"image\/jpg\" alt=\"Line graph of jet car displacement in meters versus time in seconds. The line is straight with a positive slope. The y intercept is four hundred meters. The total change in time is eight point zero seconds. The initial position is four hundred meters. The final position is two thousand meters.\" width=\"450\"><\/span><\/p><\/div>\n<p id=\"import-auto-id1761347\">Using the relationship between dependent and independent variables, we see that the slope in the graph above is average velocity [latex]\\stackrel{-}{v}[\/latex] and the intercept is displacement at time zero\u2014that is, [latex]{x}_{0}[\/latex]. Substituting these symbols into [latex]y=\\text{mx}+b[\/latex] gives<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id4019047\">[latex]x=\\stackrel{-}{v}t+{x}_{0}[\/latex]<\/div>\n<p id=\"import-auto-id2349611\">or<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2357165\">[latex]x={x}_{0}+\\stackrel{-}{v}t.[\/latex]<\/div>\n<p id=\"import-auto-id1710420\">Thus a graph of displacement versus time gives a general relationship among displacement, velocity, and time, as well as giving detailed numerical information about a specific situation. <\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id4125096\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">The Slope of <em data-effect=\"italics\">x<\/em>  vs. <em data-effect=\"italics\">t<\/em> <\/div>\n<p id=\"import-auto-id1573573\">The slope of the graph of displacement [latex]x[\/latex] vs. time [latex]t[\/latex]<em data-effect=\"italics\"> is velocity [latex]v[\/latex].<\/em><\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id4121188\">[latex]\\text{slope}=\\frac{\\Delta x}{\\Delta t}=v[\/latex]<\/div>\n<p id=\"import-auto-id1180080\">Notice that this equation is the same as that derived algebraically from other motion equations in <a href=\"\/contents\/ea2bb23c-4fce-4e9d-a46b-3754125da988@10\">Motion Equations for Constant Acceleration in One Dimension<\/a>.\n<\/p>\n<\/div>\n<p id=\"import-auto-id954017\">From the figure we can see that the car has a displacement of 25 m at 0.50 s and 2000 m at 6.40 s. Its displacement at other times can be read from the graph; furthermore, information about its velocity and acceleration can also be obtained from the graph.<\/p>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id1714610\">\n<div data-type=\"title\" class=\"title\">Determining Average Velocity from a Graph of Displacement versus Time: Jet Car<\/div>\n<p id=\"import-auto-id2092422\">Find the average velocity of the car whose position is graphed in   <a href=\"#import-auto-id2574769\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<p id=\"import-auto-id4020586\"><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id2362743\">The slope of a graph of [latex]x[\/latex] vs. [latex]t[\/latex] is average velocity, since slope equals rise over run. In this case, rise = change in displacement and run = change in time, so that<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id1850584\">[latex]\\text{slope}=\\frac{\\Delta x}{\\Delta t}=\\stackrel{-}{v}.[\/latex]<\/div>\n<p id=\"import-auto-id1666708\">Since the slope is constant here, any two points on the graph can be used to find the slope. (Generally speaking, it is most accurate to use two widely separated points on the straight line. This is because any error in reading data from the graph is proportionally smaller if the interval is larger.)<\/p>\n<p id=\"import-auto-id3539933\"><strong>Solution<\/strong><\/p>\n<p id=\"import-auto-id3577097\">1. Choose two points on the line. In this case, we choose the points labeled on the graph: (6.4 s, 2000 m) and (0.50 s, 525 m). (Note, however, that you could choose any two points.)<\/p>\n<p id=\"import-auto-id1387973\">2. Substitute the [latex]x[\/latex] and [latex]t[\/latex] values of the chosen points into the equation. Remember in calculating change [latex]\\left(\\Delta \\right)[\/latex] we always use final value minus initial value.<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id4181834\">[latex]\\stackrel{-}{v}=\\frac{\\Delta x}{\\Delta t}=\\frac{\\text{2000 m}-\\text{525 m}}{6\\text{.}\\text{4 s}-0\\text{.}\\text{50 s}},[\/latex]<\/div>\n<p id=\"import-auto-id1731744\">yielding<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2024433\">[latex]\\stackrel{-}{v}=\\text{250 m\/s}.[\/latex]<\/div>\n<p id=\"import-auto-id2589974\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id4173862\">This is an impressively large land speed (900 km\/h, or about 560 mi\/h): much greater than the typical highway speed limit of 60 mi\/h (27 m\/s or 96 km\/h), but considerably shy of the record of 343 m\/s (1234 km\/h or 766 mi\/h) set in 1997.<\/p>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id2570304\">\n<h1 data-type=\"title\">Graphs of Motion when [latex]a[\/latex] is constant but [latex]a\\ne 0[\/latex]<\/h1>\n<p id=\"import-auto-id4044849\">The graphs in <a href=\"#import-auto-id3596921\" class=\"autogenerated-content\">(Figure)<\/a> below represent the motion of the jet-powered car as it accelerates toward its top speed, but only during the time when its acceleration is constant. Time starts at zero for this motion (as if measured with a stopwatch), and the displacement and velocity are initially 200 m and 15 m\/s, respectively.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id3596921\">\n<div class=\"bc-figcaption figcaption\">Graphs of motion of a jet-powered car during the time span when its acceleration is constant. (a) The slope of an [latex]x[\/latex] vs. [latex]t[\/latex] graph is velocity. This is shown at two points, and the instantaneous velocities obtained are plotted in the next graph. Instantaneous velocity at any point is the slope of the tangent at that point. (b) The slope of the [latex]v[\/latex] vs. [latex]t[\/latex] graph is constant for this part of the motion, indicating constant acceleration. (c) Acceleration has the constant value of [latex]5\\text{.}{\\text{0 m\/s}}^{2}[\/latex] over the time interval plotted.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id3596922\" data-alt=\"Three line graphs. First is a line graph of displacement over time. Line has a positive slope that increases with time. Second line graph is of velocity over time. Line is straight with a positive slope. Third line graph is of acceleration over time. Line is straight and horizontal, indicating constant acceleration.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_02_07_03.jpg\" data-media-type=\"image\/jpg\" alt=\"Three line graphs. First is a line graph of displacement over time. Line has a positive slope that increases with time. Second line graph is of velocity over time. Line is straight with a positive slope. Third line graph is of acceleration over time. Line is straight and horizontal, indicating constant acceleration.\" width=\"300\"><\/span><\/p><\/div>\n<div class=\"bc-figure figure\" id=\"import-auto-id3583460\">\n<div class=\"bc-figcaption figcaption\">A U.S. Air Force jet car speeds down a track. (credit: Matt Trostle, Flickr)<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id3583462\" data-alt=\"\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_02_07_03a.jpg\" data-media-type=\"image\/png\" alt=\"\" width=\"300\"><\/span><\/p><\/div>\n<p id=\"import-auto-id2379100\">The graph of displacement versus time in <a href=\"#import-auto-id3596921\" class=\"autogenerated-content\">(Figure)<\/a>(a) is a curve rather than a straight line. The slope of the curve becomes steeper as time progresses, showing that the velocity is increasing over time. The slope at any point on a displacement-versus-time graph is the instantaneous velocity at that point. It is found by drawing a straight line tangent to the curve at the point of interest and taking the slope of this straight line. Tangent lines are shown for two points in <a href=\"#import-auto-id3596921\" class=\"autogenerated-content\">(Figure)<\/a>(a). If this is done at every point on the curve and the values are plotted against time, then the graph of velocity versus time shown in <a href=\"#import-auto-id3596921\" class=\"autogenerated-content\">(Figure)<\/a>(b) is obtained. Furthermore, the slope of the graph of velocity versus time is acceleration, which is shown in <a href=\"#import-auto-id3596921\" class=\"autogenerated-content\">(Figure)<\/a>(c).<\/p>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id1516659\">\n<div data-type=\"title\" class=\"title\">Determining Instantaneous Velocity from the Slope at a Point: Jet Car<\/div>\n<p id=\"import-auto-id1752646\">Calculate the velocity of the jet car at a time of 25 s by finding the slope of the [latex]x[\/latex] vs. [latex]t[\/latex] graph in the graph below.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id4141386\">\n<div class=\"bc-figcaption figcaption\">The slope of an [latex]x[\/latex] vs. [latex]t[\/latex] graph is velocity. This is shown at two points. Instantaneous velocity at any point is the slope of the tangent at that point.\n      <\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1550759\" data-alt=\"A graph of displacement versus time for a jet car. The x axis for time runs from zero to thirty five seconds. The y axis for displacement runs from zero to three thousand meters. The curve depicting displacement is concave up. The slope of the curve increases over time. Slope equals velocity v. There are two points on the curve, labeled, P and Q. P is located at time equals ten seconds. Q is located and time equals twenty-five seconds. A line tangent to P at ten seconds is drawn and has a slope delta x sub P over delta t sub p. A line tangent to Q at twenty five seconds is drawn and has a slope equal to delta x sub q over delta t sub q. Select coordinates are given in a table and consist of the following: time zero seconds displacement two hundred meters; time five seconds displacement three hundred thirty eight meters; time ten seconds displacement six hundred meters; time fifteen seconds displacement nine hundred eighty eight meters. Time twenty seconds displacement one thousand five hundred meters; time twenty five seconds displacement two thousand one hundred thirty eight meters; time thirty seconds displacement two thousand nine hundred meters.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/graphics4-1.jpg\" data-media-type=\"image\/jpg\" alt=\"A graph of displacement versus time for a jet car. The x axis for time runs from zero to thirty five seconds. The y axis for displacement runs from zero to three thousand meters. The curve depicting displacement is concave up. The slope of the curve increases over time. Slope equals velocity v. There are two points on the curve, labeled, P and Q. P is located at time equals ten seconds. Q is located and time equals twenty-five seconds. A line tangent to P at ten seconds is drawn and has a slope delta x sub P over delta t sub p. A line tangent to Q at twenty five seconds is drawn and has a slope equal to delta x sub q over delta t sub q. Select coordinates are given in a table and consist of the following: time zero seconds displacement two hundred meters; time five seconds displacement three hundred thirty eight meters; time ten seconds displacement six hundred meters; time fifteen seconds displacement nine hundred eighty eight meters. Time twenty seconds displacement one thousand five hundred meters; time twenty five seconds displacement two thousand one hundred thirty eight meters; time thirty seconds displacement two thousand nine hundred meters.\" width=\"300\"><\/span><\/p><\/div>\n<p id=\"import-auto-id4041504\"><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id2333254\">The slope of a curve at a point is equal to the slope of a straight line tangent to the curve at that point. This principle is illustrated in <a href=\"#import-auto-id4141386\" class=\"autogenerated-content\">(Figure)<\/a>, where Q is the point at [latex]t=\\text{25 s}[\/latex].<\/p>\n<p id=\"import-auto-id1729462\"><strong>Solution<\/strong><\/p>\n<p id=\"import-auto-id4063815\">1. Find the tangent line to the curve at [latex]t=\\text{25 s}[\/latex]. <\/p>\n<p id=\"import-auto-id3531643\">2. Determine the endpoints of the tangent. These correspond to a position of 1300 m at time 19 s and a position of 3120 m at time 32 s.<\/p>\n<p id=\"import-auto-id2034293\">3. Plug these endpoints into the equation to solve for the slope, <em data-effect=\"italics\">[latex]v[\/latex]<\/em>. <\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id3627758\">[latex]\\text{slope}={v}_{Q}=\\frac{{\\Delta x}_{Q}}{{\\Delta t}_{Q}}=\\frac{\\left(\\text{3120 m}-\\text{1300 m}\\right)}{\\left(\\text{32 s}-\\text{19 s}\\right)}[\/latex]<\/div>\n<p id=\"import-auto-id2325601\">Thus,<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id1657105\">[latex]{v}_{Q}=\\frac{\\text{1820 m}}{\\text{13 s}}=\\text{140 m\/s.}[\/latex]<\/div>\n<p id=\"import-auto-id2364077\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id4020469\">This is the value given in this figure\u2019s table for [latex]v[\/latex] at [latex]t=\\text{25 s}[\/latex]. The value of 140 m\/s for [latex]{v}_{Q}[\/latex] is plotted in <a href=\"#import-auto-id4141386\" class=\"autogenerated-content\">(Figure)<\/a>. The entire graph of [latex]v[\/latex] vs. [latex]t[\/latex] can be obtained in this fashion.<\/p>\n<\/div>\n<p id=\"import-auto-id1690009\">Carrying this one step further, we note that the slope of a velocity versus time graph is acceleration. Slope is rise divided by run; on a [latex]v[\/latex] vs. [latex]t[\/latex] graph, rise = change in velocity [latex]\\Delta v[\/latex] and run = change in time [latex]\\Delta t[\/latex].<\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id1405001\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">The Slope of <em data-effect=\"italics\">v<\/em> vs. <em data-effect=\"italics\">t<\/em><\/div>\n<p id=\"import-auto-id737611\">The slope of a graph of velocity [latex]v[\/latex] vs. time [latex]t[\/latex] is acceleration [latex]a[\/latex]<em data-effect=\"italics\">.<\/em><\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id4096826\">[latex]\\text{slope}=\\frac{\\Delta v}{\\Delta t}=a[\/latex]<\/div>\n<\/div>\n<p id=\"import-auto-id4021093\">Since the velocity versus time graph in <a href=\"#import-auto-id3596921\" class=\"autogenerated-content\">(Figure)<\/a>(b) is a straight line, its slope is the same everywhere, implying that acceleration is constant. Acceleration versus time is graphed in <a href=\"#import-auto-id3596921\" class=\"autogenerated-content\">(Figure)<\/a>(c).<\/p>\n<p id=\"import-auto-id4073634\">Additional general information can be obtained from <a href=\"#import-auto-id4141386\" class=\"autogenerated-content\">(Figure)<\/a> and the expression for a straight line, [latex]y=\\text{mx}+b[\/latex].<\/p>\n<p>In this case, the vertical axis [latex]y[\/latex] is [latex]V[\/latex], the intercept [latex]b[\/latex] is [latex]{v}_{0}[\/latex], the slope [latex]m[\/latex] is [latex]a[\/latex], and the horizontal axis [latex]x[\/latex] is [latex]t[\/latex]. Substituting these symbols yields<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id1714581\">[latex]v={v}_{0}+\\text{at}.[\/latex]<\/div>\n<p id=\"import-auto-id1763694\">A general relationship for velocity, acceleration, and time has again been obtained from a graph. Notice that this equation was also derived algebraically from other motion equations in <a href=\"\/contents\/ea2bb23c-4fce-4e9d-a46b-3754125da988@10\">Motion Equations for Constant Acceleration in One Dimension<\/a>.<\/p>\n<p id=\"import-auto-id4171982\">It is not accidental that the same equations are obtained by graphical analysis as by algebraic techniques. In fact, an important way to <em data-effect=\"italics\">discover<\/em> physical relationships is to measure various physical quantities and then make graphs of one quantity against another to see if they are correlated in any way. Correlations imply physical relationships and might be shown by smooth graphs such as those above. From such graphs, mathematical relationships can sometimes be postulated. Further experiments are then performed to determine the validity of the hypothesized relationships.<\/p>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id2306208\">\n<h1 data-type=\"title\">Graphs of Motion Where Acceleration is Not Constant<\/h1>\n<p id=\"import-auto-id1544762\">Now consider the motion of the jet car as it goes from 165 m\/s to its top velocity of 250 m\/s, graphed in <a href=\"#import-auto-id1534076\" class=\"autogenerated-content\">(Figure)<\/a>. Time again starts at zero, and the initial displacement and velocity are 2900 m and 165 m\/s, respectively. (These were the final displacement and velocity of the car in the motion graphed in <a href=\"#import-auto-id3596921\" class=\"autogenerated-content\">(Figure)<\/a>.) Acceleration gradually decreases from [latex]5\\text{.}{\\text{0 m\/s}}^{2}[\/latex] to zero when the car hits 250 m\/s. The slope of the [latex]x[\/latex] vs. [latex]t[\/latex] graph increases until [latex]t=\\text{55 s}[\/latex], after which time the slope is constant. Similarly, velocity increases until 55 s and then becomes constant, since acceleration decreases to zero at 55 s and remains zero afterward.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1534076\">\n<div class=\"bc-figcaption figcaption\">Graphs of motion of a jet-powered car as it reaches its top velocity. This motion begins where the motion in <a href=\"#import-auto-id3596921\" class=\"autogenerated-content\">(Figure)<\/a> ends. (a) The slope of this graph is velocity; it is plotted in the next graph. (b) The velocity gradually approaches its top value. The slope of this graph is acceleration; it is plotted in the final graph. (c) Acceleration gradually declines to zero when velocity becomes constant.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1534078\" data-alt=\"Three line graphs of jet car displacement, velocity, and acceleration, respectively. First line graph is of position over time. Line is straight with a positive slope. Second line graph is of velocity over time. Line graph has a positive slope that decreases over time and flattens out at the end. Third line graph is of acceleration over time. Line has a negative slope that increases over time until it flattens out at the end. The line is not smooth, but has several kinks.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_02_07_04.jpg\" data-media-type=\"image\/jpg\" alt=\"Three line graphs of jet car displacement, velocity, and acceleration, respectively. First line graph is of position over time. Line is straight with a positive slope. Second line graph is of velocity over time. Line graph has a positive slope that decreases over time and flattens out at the end. Third line graph is of acceleration over time. Line has a negative slope that increases over time until it flattens out at the end. The line is not smooth, but has several kinks.\" width=\"350\"><\/span><\/p><\/div>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id1406638\">\n<div data-type=\"title\" class=\"title\">Calculating Acceleration from a Graph of Velocity versus Time<\/div>\n<p id=\"import-auto-id1364942\">Calculate the acceleration of the jet car at a time of 25 s by finding the slope of the [latex]v[\/latex] vs. [latex]t[\/latex] graph in <a href=\"#import-auto-id1534076\" class=\"autogenerated-content\">(Figure)<\/a>(b).<\/p>\n<p id=\"import-auto-id3600663\"><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id2165878\">The slope of the curve at [latex]t=\\text{25 s}[\/latex] is equal to the slope of the line tangent at that point, as illustrated in <a href=\"#import-auto-id1534076\" class=\"autogenerated-content\">(Figure)<\/a>(b).<\/p>\n<p id=\"import-auto-id945644\"><strong>Solution<\/strong><\/p>\n<p id=\"import-auto-id4081606\">Determine endpoints of the tangent line from the figure, and then plug them into the equation to solve for slope, [latex]a[\/latex].<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id3503054\">[latex]\\text{slope}=\\frac{\\Delta v}{\\Delta t}=\\frac{\\left(\\text{260 m\/s}-\\text{210 m\/s}\\right)}{\\left(\\text{51 s}-1.0 s\\right)}[\/latex]<\/div>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2028886\">[latex]a=\\frac{\\text{50 m\/s}}{\\text{50 s}}=1\\text{.}0 m{\\text{\/s}}^{2}.[\/latex]<\/div>\n<p id=\"import-auto-id1568664\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id2186238\">Note that this value for [latex]a[\/latex] is consistent with the value plotted in <a href=\"#import-auto-id1534076\" class=\"autogenerated-content\">(Figure)<\/a>(c) at [latex]t=\\text{25 s}[\/latex].<\/p>\n<\/div>\n<p id=\"eip-788\">A graph of displacement versus time can be used to generate a graph of velocity versus time, and a graph of velocity versus time can be used to generate a graph of acceleration versus time. We do this by finding the slope of the graphs at every point. If the graph is linear (i.e., a line with a constant slope), it is easy to find the slope at any point and you have the slope for every point. Graphical analysis of motion can be used to describe both specific and general characteristics of kinematics. Graphs can also be used for other topics in physics. An important aspect of exploring physical relationships is to graph them and look for underlying relationships.<\/p>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1571006\" 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-id1429801\">\n<p id=\"import-auto-id2305974\">A graph of velocity vs. time of a ship coming into a harbor is shown below. (a) Describe the motion of the ship based on the graph. (b)What would a graph of the ship\u2019s acceleration look like? <\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id3504346\"><span data-type=\"media\" id=\"import-auto-id1957571\" data-alt=\"Line graph of velocity versus time. The line has three legs. The first leg is flat. The second leg has a negative slope. The third leg also has a negative slope, but the slope is not as negative as the second leg.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_02_07_04a.jpg\" data-media-type=\"image\/jpg\" alt=\"Line graph of velocity versus time. The line has three legs. The first leg is flat. The second leg has a negative slope. The third leg also has a negative slope, but the slope is not as negative as the second leg.\" width=\"200\"><\/span><\/div>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1658952\">\n<p id=\"import-auto-id4086286\">(a) The ship moves at constant velocity and then begins to decelerate at a constant rate. At some point, its deceleration rate decreases. It maintains this lower deceleration rate until it stops moving.<\/p>\n<p id=\"import-auto-id1850777\">(b) A graph of acceleration vs. time would show zero acceleration in the first leg, large and constant negative acceleration in the second leg, and constant negative acceleration.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1666671\"><span data-type=\"media\" id=\"import-auto-id1666672\" data-alt=\"A line graph of acceleration versus time. There are three legs of the graph. All three legs are flat and straight. The first leg shows constant acceleration of 0. The second leg shows a constant negative acceleration. The third leg shows a constant negative acceleration that is not as negative as the second leg.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_02_07_04b.jpg\" data-media-type=\"image\/jpg\" alt=\"A line graph of acceleration versus time. There are three legs of the graph. All three legs are flat and straight. The first leg shows constant acceleration of 0. The second leg shows a constant negative acceleration. The third leg shows a constant negative acceleration that is not as negative as the second leg.\" width=\"200\"><\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"section-summary\" data-depth=\"1\" id=\"fs-id1762928\">\n<h1 data-type=\"title\">Section Summary<\/h1>\n<ul id=\"eip-id4070172\">\n<li id=\"import-auto-id2388505\">Graphs of motion can be used to analyze motion. <\/li>\n<li id=\"import-auto-id4097898\">Graphical solutions yield identical solutions to mathematical methods for deriving motion equations.<\/li>\n<li id=\"import-auto-id2294483\">The slope of a graph of displacement [latex]x[\/latex] vs. time [latex]t[\/latex] is velocity [latex]v[\/latex]<em data-effect=\"italics\">.<\/em><\/li>\n<li id=\"import-auto-id2025741\">The slope of a graph of velocity [latex]v[\/latex]<em data-effect=\"italics\"> vs. time [latex]t[\/latex] graph is acceleration [latex]a[\/latex]<em data-effect=\"italics\">.<\/em><\/em><\/li>\n<li id=\"import-auto-id1561758\">Average velocity, instantaneous velocity, and acceleration can all be obtained by analyzing graphs.<\/li>\n<\/ul>\n<\/div>\n<div class=\"conceptual-questions\" data-depth=\"1\" data-element-type=\"conceptual-questions\">\n<h1 data-type=\"title\">Conceptual Questions<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1550042\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id4182585\">\n<p id=\"import-auto-id3510347\">(a) Explain how you can use the graph of position versus time in <a href=\"#import-auto-id4064025\" class=\"autogenerated-content\">(Figure)<\/a> to describe the change in velocity over time. Identify (b) the time ([latex]{t}_{a}[\/latex],<br>\n[latex]{t}_{b}[\/latex],<br>\n[latex]{t}_{c}[\/latex],<br>\n[latex]{t}_{d}[\/latex], or<br>\n[latex]{t}_{e}[\/latex]) at which the instantaneous velocity is greatest, (c) the time at which it is zero, and (d) the time at which it is negative.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id4064025\"><span data-type=\"media\" id=\"import-auto-id4033064\" data-alt=\"Line graph of position versus time with 5 points labeled: a, b, c, d, and e. The slope of the line changes. It begins with a positive slope that decreases over time until around point d, where it is flat. It then has a slightly negative slope.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_08Sol_01.jpg\" data-media-type=\"image\/jpg\" alt=\"Line graph of position versus time with 5 points labeled: a, b, c, d, and e. The slope of the line changes. It begins with a positive slope that decreases over time until around point d, where it is flat. It then has a slightly negative slope.\" width=\"300\"><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id4168594\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1759884\">\n<p id=\"import-auto-id1784242\">(a) Sketch a graph of velocity versus time corresponding to the graph of displacement versus time given in <a href=\"#import-auto-id2562897\" class=\"autogenerated-content\">(Figure)<\/a>. (b) Identify the time or times ([latex]{t}_{a}[\/latex],<br>\n[latex]{t}_{b}[\/latex],<br>\n[latex]{t}_{c}[\/latex], etc.) at which the instantaneous velocity is greatest. (c) At which times is it zero? (d) At which times is it negative?<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id2562897\"><span data-type=\"media\" id=\"import-auto-id2562898\" data-alt=\"Line graph of position over time with 12 points labeled a through l. Line has a negative slope from a to c, where it turns and has a positive slope till point e. It turns again and has a negative slope till point g. The slope then increases again till l, where it flattens out.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_08Sol_02.jpg\" data-media-type=\"image\/jpg\" alt=\"Line graph of position over time with 12 points labeled a through l. Line has a negative slope from a to c, where it turns and has a positive slope till point e. It turns again and has a negative slope till point g. The slope then increases again till l, where it flattens out.\" width=\"300\"><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1549493\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1549495\">\n<p id=\"import-auto-id1778974\">(a) Explain how you can determine the acceleration over time from a velocity versus time graph such as the one in <a href=\"#import-auto-id1778975\" class=\"autogenerated-content\">(Figure)<\/a>. (b) Based on the graph, how does acceleration change over time?<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1778975\"><span data-type=\"media\" id=\"import-auto-id4083120\" data-alt=\"Line graph of velocity over time with two points labeled. Point P is at v 1 t 1. Point Q is at v 2 t 2. The line has a positive slope that increases over time.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_08Sol_04.jpg\" data-media-type=\"image\/jpg\" alt=\"Line graph of velocity over time with two points labeled. Point P is at v 1 t 1. Point Q is at v 2 t 2. The line has a positive slope that increases over time.\" width=\"300\"><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id4131202\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2086598\">\n<p id=\"import-auto-id4033189\">(a) Sketch a graph of acceleration versus time corresponding to the graph of velocity versus time given in <a href=\"#import-auto-id1447833\" class=\"autogenerated-content\">(Figure)<\/a>. (b) Identify the time or times ([latex]{t}_{a}[\/latex],<br>\n[latex]{t}_{b}[\/latex],<br>\n[latex]{t}_{c}[\/latex], etc.) at which the acceleration is greatest. (c) At which times is it zero? (d) At which times is it negative?<\/p>\n<p id=\"import-auto-id1447832\">\n<\/p><div class=\"bc-figure figure\" id=\"import-auto-id1447833\"><span data-type=\"media\" id=\"import-auto-id1447834\" data-alt=\"Line graph of velocity over time with 12 points labeled a through l. The line has a positive slope from a at the origin to d where it slopes downward to e, and then back upward to h. It then slopes back down to point l at v equals 0.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_08Sol_05.jpg\" data-media-type=\"image\/jpg\" alt=\"Line graph of velocity over time with 12 points labeled a through l. The line has a positive slope from a at the origin to d where it slopes downward to e, and then back upward to h. It then slopes back down to point l at v equals 0.\" width=\"300\"><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1365827\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id4021330\">\n<p id=\"import-auto-id4047633\">Consider the velocity vs. time graph of a person in an elevator shown in <a href=\"#import-auto-id2006890\" class=\"autogenerated-content\">(Figure)<\/a>. Suppose the elevator is initially at rest. It then accelerates for 3 seconds, maintains that velocity for 15 seconds, then decelerates for 5 seconds until it stops. The acceleration for the entire trip is not constant so we cannot use the equations of motion from <a href=\"\/contents\/ea2bb23c-4fce-4e9d-a46b-3754125da988@10\">Motion Equations for Constant Acceleration in One Dimension<\/a> for the complete trip. (We could, however, use them in the three individual sections where acceleration is a constant.) Sketch graphs of (a) position vs. time and (b) acceleration vs. time for this trip.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id2006890\"><span data-type=\"media\" id=\"import-auto-id2006892\" data-alt=\"Line graph of velocity versus time. Line begins at the origin and has a positive slope until it reaches 3 meters per second at 3 seconds. The slope is then zero until 18 seconds, where it becomes negative until the line reaches a velocity of 0 at 23 seconds.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_08Sol_07.jpg\" data-media-type=\"image\/jpg\" alt=\"Line graph of velocity versus time. Line begins at the origin and has a positive slope until it reaches 3 meters per second at 3 seconds. The slope is then zero until 18 seconds, where it becomes negative until the line reaches a velocity of 0 at 23 seconds.\" width=\"350\"><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2576953\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2589937\">\n<p id=\"import-auto-id4124847\">A cylinder is given a push and then rolls up an inclined plane. If the origin is the starting point, sketch the position, velocity, and acceleration of the cylinder vs. time as it goes up and then down the plane. <\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"problems-exercises\" data-depth=\"1\" id=\"fs-id1987308\" data-element-type=\"problems-exercises\">\n<h1 data-type=\"title\">Problems &amp; Exercises<\/h1>\n<p id=\"import-auto-id1960253\">Note: There is always uncertainty in numbers taken from graphs. If your answers differ from expected values, examine them to see if they are within data extraction uncertainties estimated by you.<\/p>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id4088406\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id4088408\">\n<p id=\"import-auto-id1729859\">(a) By taking the slope of the curve in <a href=\"#import-auto-id1798398\" class=\"autogenerated-content\">(Figure)<\/a>, verify that the velocity of the jet car is 115 m\/s at [latex]t=\\text{20 s}[\/latex]. (b) By taking the slope of the curve at any point in <a href=\"#import-auto-id4101417\" class=\"autogenerated-content\">(Figure)<\/a>, verify that the jet car\u2019s acceleration is [latex]5\\text{.}{\\text{0 m\/s}}^{2}[\/latex].<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1798398\"><span data-type=\"media\" id=\"import-auto-id4028860\" data-alt=\"Line graph of position over time. Line has positive slope that increases over time.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_02_08Sol_11.jpg\" data-media-type=\"image\/jpg\" alt=\"Line graph of position over time. Line has positive slope that increases over time.\" width=\"350\"><\/span><\/div>\n<div class=\"bc-figure figure\" id=\"import-auto-id4101417\"><span data-type=\"media\" id=\"import-auto-id4101418\" data-alt=\"Line graph of velocity versus time. Line is straight with a positive slope.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_02_08Sol_12.jpg\" data-media-type=\"image\/jpg\" alt=\"Line graph of velocity versus time. Line is straight with a positive slope.\" width=\"350\"><\/span><\/div>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2295253\">\n<p id=\"import-auto-id952467\">(a) [latex]\\text{115 m\/s}[\/latex]<\/p>\n<p id=\"import-auto-id1510856\">(b) [latex]5\\text{.}{\\text{0 m\/s}}^{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id4012994\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id4012996\">\n<p id=\"import-auto-id2006034\">Using approximate values, calculate the slope of the curve in <a href=\"#import-auto-id4122996\" class=\"autogenerated-content\">(Figure)<\/a> to verify that the velocity at [latex]t=\\text{10.0 s}[\/latex] is 0.208 m\/s. Assume all values are known to 3 significant figures.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id4122996\"><span data-type=\"media\" id=\"import-auto-id2015268\" data-alt=\"Line graph of position versus time. Line is straight with a positive slope.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_02_08Sol_13.jpg\" data-media-type=\"image\/jpg\" alt=\"Line graph of position versus time. Line is straight with a positive slope.\" width=\"350\"><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1770908\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1770911\">\n<p id=\"import-auto-id1761648\">Using approximate values, calculate the slope of the curve in <a href=\"#import-auto-id4122996\" class=\"autogenerated-content\">(Figure)<\/a> to verify that the velocity at [latex]t=\\text{30.0 s}[\/latex] is 0.238 m\/s. Assume all values are known to 3 significant figures.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1707522\">\n<div data-type=\"equation\" class=\"equation\" id=\"eip-id2453534\">[latex]v=\\frac{\\left(\\text{11.7}-6.95\\right)\u00d7{\\text{10}}^{3}\\phantom{\\rule{0.25em}{0ex}}\\text{m}}{\\left(40\\text{.}\\text{0 \u2013 20}.0\\right)\\phantom{\\rule{0.25em}{0ex}}\\text{s}}=\\text{238 m\/s}[\/latex]<\/div>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2475925\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1744756\">\n<p id=\"import-auto-id1743717\">By taking the slope of the curve in <a href=\"#import-auto-id3552017\" class=\"autogenerated-content\">(Figure)<\/a>, verify that the acceleration is [latex]3\\text{.}2 m{\\text{\/s}}^{2}[\/latex] at [latex]t=\\text{10 s}[\/latex].<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id3552017\"><span data-type=\"media\" id=\"import-auto-id3552018\" data-alt=\"Line graph of velocity versus time. Line has a positive slope that decreases over time until the line flattens out.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_02_08Sol_14.jpg\" data-media-type=\"image\/jpg\" alt=\"Line graph of velocity versus time. Line has a positive slope that decreases over time until the line flattens out.\" width=\"350\"><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1372323\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1544965\">\n<p id=\"import-auto-id3575253\">Construct the displacement graph for the subway shuttle train as shown in <a href=\"\/contents\/6023b87d-5a28-4910-9e51-ee7fd11c98e1@4#import-auto-id2590556\" class=\"autogenerated-content\">(Figure)<\/a>(a). Your graph should show the position of the train, in kilometers, from t = 0 to 20 s. You will need to use the information on acceleration and velocity given in the examples for this figure. <\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1778988\">\n<div class=\"bc-figure figure\" id=\"import-auto-id3597350\"><span data-type=\"media\" id=\"import-auto-id3597352\" data-alt=\"Line graph of position versus time. Line begins with a slight positive slope. It then kinks to a much greater positive slope.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_02_08Sol_15.jpg\" data-media-type=\"image\/jpg\" alt=\"Line graph of position versus time. Line begins with a slight positive slope. It then kinks to a much greater positive slope.\" width=\"350\"><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2290187\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2290189\">\n<p id=\"import-auto-id2592264\">(a) Take the slope of the curve in <a href=\"#import-auto-id4064858\" class=\"autogenerated-content\">(Figure)<\/a> to find the jogger\u2019s velocity at [latex]t=2\\text{.}5 s[\/latex]. (b) Repeat at 7.5 s. These values must be consistent with the graph in <a href=\"#import-auto-id4128350\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id4064858\"><span data-type=\"media\" id=\"import-auto-id4064859\" data-alt=\"Line graph of position over time. Line begins sloping upward, then kinks back down, then kinks back upward again.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_02_08Sol_16.jpg\" data-media-type=\"image\/jpg\" alt=\"Line graph of position over time. Line begins sloping upward, then kinks back down, then kinks back upward again.\" width=\"300\"><\/span><\/div>\n<div class=\"bc-figure figure\" id=\"import-auto-id4128350\"><span data-type=\"media\" id=\"import-auto-id4128351\" data-alt=\"Line graph of velocity over time. Line begins with a positive slope, then kinks downward with a negative slope, then kinks back upward again. It kinks back down again slightly, then back up again, and ends with a slightly less positive slope.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_02_08Sol_17.jpg\" data-media-type=\"image\/jpg\" alt=\"Line graph of velocity over time. Line begins with a positive slope, then kinks downward with a negative slope, then kinks back upward again. It kinks back down again slightly, then back up again, and ends with a slightly less positive slope.\" width=\"300\"><\/span><\/div>\n<div class=\"bc-figure figure\" id=\"import-auto-id4151339\"><span data-type=\"media\" id=\"import-auto-id4151340\" data-alt=\"\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_02_08Sol_18.jpg\" data-media-type=\"image\/jpg\" alt=\"\" width=\"300\"><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3520768\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1434602\">\n<p id=\"import-auto-id1820030\">A graph of [latex]v\\left(t\\right)[\/latex] is shown for a world-class track sprinter in a 100-m race. (See  <a href=\"#import-auto-id4125036\" class=\"autogenerated-content\">(Figure)<\/a>). (a) What is his average velocity for the first 4 s? (b) What is his instantaneous velocity at [latex]t=5 s[\/latex]? (c) What is his average acceleration between 0 and 4 s? (d) What is his time for the race?<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id4125036\"><span data-type=\"media\" id=\"import-auto-id4125037\" data-alt=\"Line graph of velocity versus time. The line has two legs. The first has a constant positive slope. The second is flat, with a slope of 0.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_02_08Sol_20.jpg\" data-media-type=\"image\/jpg\" alt=\"Line graph of velocity versus time. The line has two legs. The first has a constant positive slope. The second is flat, with a slope of 0.\" width=\"350\"><\/span><\/div>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1778256\">\n<p id=\"import-auto-id1746146\">(a) 6 m\/s<\/p>\n<p id=\"import-auto-id1746150\">(b) 12 m\/s<\/p>\n<p id=\"import-auto-id3624887\">(c) [latex]{\\text{3 m\/s}}^{2}[\/latex]<\/p>\n<p id=\"import-auto-id1757765\">(d) 10 s<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1582774\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1598940\">\n<p id=\"import-auto-id1730106\"><a href=\"#import-auto-id4035681\" class=\"autogenerated-content\">(Figure)<\/a> shows the displacement graph for a particle for 5 s. Draw the corresponding velocity and acceleration graphs. <\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id4035681\"><span data-type=\"media\" id=\"import-auto-id4035682\" data-alt=\"Line graph of position versus time. The line has 4 legs. The first leg has a positive slope. The second leg has a negative slope. The third has a slope of 0. The fourth has a positive slope.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_02_08Sol_21.jpg\" data-media-type=\"image\/jpg\" alt=\"Line graph of position versus time. The line has 4 legs. The first leg has a positive slope. The second leg has a negative slope. The third has a slope of 0. The fourth has a positive slope.\" width=\"350\"><\/span><\/div>\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=\"fs-id1218535\">\n<dt>independent variable<\/dt>\n<dd id=\"fs-id1343248\">the variable that the dependent variable is measured with respect to; usually plotted along the [latex]x[\/latex]-axis\n<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id1845390\">\n<dt>dependent variable<\/dt>\n<dd id=\"fs-id4015576\">the variable that is being measured; usually plotted along the [latex]y[\/latex]-axis\n<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id3600469\">\n<dt>slope<\/dt>\n<dd id=\"fs-id1544972\">the difference in [latex]y[\/latex]-value (the rise) divided by the difference in [latex]x[\/latex]-value (the run) of two points on a straight line<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id4021637\">\n<dt>y-intercept<\/dt>\n<dd id=\"fs-id3525350\">the [latex]y\\text{-}[\/latex]value when [latex]x[\/latex]<em data-effect=\"italics\">= 0, or when the graph crosses the [latex]y[\/latex]-axis<\/em><\/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 a straight-line graph in terms of its slope and <span data-type=\"foreign\">y<\/span>-intercept.<\/li>\n<li>Determine average velocity or instantaneous velocity from a graph of position vs. time.<\/li>\n<li>Determine average or instantaneous acceleration from a graph of velocity vs. time.<\/li>\n<li>Derive a graph of velocity vs. time from a graph of position vs. time.<\/li>\n<li>Derive a graph of acceleration vs. time from a graph of velocity vs. time.<\/li>\n<\/ul>\n<\/div>\n<p id=\"import-auto-id1568723\">A graph, like a picture, is worth a thousand words. Graphs not only contain numerical information; they also reveal relationships between physical quantities. This section uses graphs of displacement, velocity, and acceleration versus time to illustrate one-dimensional kinematics.<\/p>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1396690\">\n<h1 data-type=\"title\">Slopes and General Relationships<\/h1>\n<p id=\"import-auto-id4131770\">First note that graphs in this text have perpendicular axes, one horizontal and the other vertical. When two physical quantities are plotted against one another in such a graph, the horizontal axis is usually considered to be an <span data-type=\"term\" id=\"import-auto-id1690042\">independent variable<\/span> and the vertical axis a <span data-type=\"term\" id=\"import-auto-id2013112\">dependent variable<\/span>. If we call the horizontal axis the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/>-axis and the vertical axis the <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;\" \/>-axis, as in <a href=\"#import-auto-id2359358\" class=\"autogenerated-content\">(Figure)<\/a>, a straight-line graph has the general form<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id4175150\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1f5d1e6a15c9e3efd82b1799c30f1192_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#120;&#125;&#43;&#98;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"91\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"import-auto-id1433182\">Here <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6b41df788161942c6f98604d37de8098_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"15\" style=\"vertical-align: 0px;\" \/> is the <span data-type=\"term\" id=\"import-auto-id1773074\">slope<\/span>, defined to be the rise divided by the run (as seen in the figure) of the straight line. The letter <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f56d50c26583f9a035ff6b4e3c0ca5c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"8\" style=\"vertical-align: 0px;\" \/> is used for the <span data-type=\"term\" id=\"import-auto-id2955215\"><em data-effect=\"italics\">y<\/em>-intercept<\/span>, which is the point at which the line crosses the vertical axis.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id2359358\">\n<div class=\"bc-figcaption figcaption\">A straight-line graph. The equation for a straight line is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3da43b1120b85198243b1ee39591d4ed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#120;&#125;&#43;&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"87\" style=\"vertical-align: -4px;\" \/><br \/>\n    .<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1534567\" data-alt=\"Graph of a straight-line sloping up at about 40 degrees.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_02_07_01.jpg\" data-media-type=\"image\/jpg\" alt=\"Graph of a straight-line sloping up at about 40 degrees.\" width=\"300\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id2201114\">\n<h1 data-type=\"title\">Graph of Displacement vs. Time (<em data-effect=\"italics\">a<\/em> = 0, so <em data-effect=\"italics\">v<\/em> is constant)<\/h1>\n<p id=\"import-auto-id1768816\">Time is usually an independent variable that other quantities, such as displacement, depend upon. A graph of displacement versus time would, thus, have <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;\" \/> on the vertical axis and <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;\" \/> on the horizontal axis.   <a href=\"#import-auto-id2574769\" class=\"autogenerated-content\">(Figure)<\/a> is just such a straight-line graph. It shows a graph of displacement versus time for a jet-powered car on a very flat dry lake bed in Nevada.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id2574769\">\n<div class=\"bc-figcaption figcaption\">Graph of displacement versus time for a jet-powered car on the Bonneville Salt Flats. <\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1599004\" data-alt=\"Line graph of jet car displacement in meters versus time in seconds. The line is straight with a positive slope. The y intercept is four hundred meters. The total change in time is eight point zero seconds. The initial position is four hundred meters. The final position is two thousand meters.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_02_07_02.jpg\" data-media-type=\"image\/jpg\" alt=\"Line graph of jet car displacement in meters versus time in seconds. The line is straight with a positive slope. The y intercept is four hundred meters. The total change in time is eight point zero seconds. The initial position is four hundred meters. The final position is two thousand meters.\" width=\"450\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id1761347\">Using the relationship between dependent and independent variables, we see that the slope in the graph above is average velocity <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b1204f2cbe796f0c0dadaef5ef056ed0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#45;&#125;&#123;&#118;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"9\" style=\"vertical-align: 1px;\" \/> and the intercept is displacement at time zero\u2014that is, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-96b224e50cd20bf6c24005d45c5a085c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"17\" style=\"vertical-align: -3px;\" \/>. Substituting these symbols into <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3da43b1120b85198243b1ee39591d4ed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#120;&#125;&#43;&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"87\" style=\"vertical-align: -4px;\" \/> gives<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id4019047\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6c370a1b0aeb46d45a0238633aeebaaa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#45;&#125;&#123;&#118;&#125;&#116;&#43;&#123;&#120;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"90\" style=\"vertical-align: -3px;\" \/><\/div>\n<p id=\"import-auto-id2349611\">or<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2357165\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a37ae32548374b0aaf4d09bed6269285_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#123;&#120;&#125;&#95;&#123;&#48;&#125;&#43;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#45;&#125;&#123;&#118;&#125;&#116;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"96\" style=\"vertical-align: -3px;\" \/><\/div>\n<p id=\"import-auto-id1710420\">Thus a graph of displacement versus time gives a general relationship among displacement, velocity, and time, as well as giving detailed numerical information about a specific situation. <\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id4125096\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">The Slope of <em data-effect=\"italics\">x<\/em>  vs. <em data-effect=\"italics\">t<\/em> <\/div>\n<p id=\"import-auto-id1573573\">The slope of the graph of displacement <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;\" \/> vs. time <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b4e3cbf5d4c5c6d9b702dd139f14c147_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"6\" style=\"vertical-align: 0px;\" \/><em data-effect=\"italics\"> is velocity <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ef71511c70f0e4b25cc6bd69f3bc20c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/>.<\/em><\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id4121188\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a56e20d85e7532e5df50c65f2e7c7aab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#108;&#111;&#112;&#101;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#120;&#125;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#116;&#125;&#61;&#118;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"118\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id1180080\">Notice that this equation is the same as that derived algebraically from other motion equations in <a href=\"\/contents\/ea2bb23c-4fce-4e9d-a46b-3754125da988@10\">Motion Equations for Constant Acceleration in One Dimension<\/a>.\n<\/p>\n<\/div>\n<p id=\"import-auto-id954017\">From the figure we can see that the car has a displacement of 25 m at 0.50 s and 2000 m at 6.40 s. Its displacement at other times can be read from the graph; furthermore, information about its velocity and acceleration can also be obtained from the graph.<\/p>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id1714610\">\n<div data-type=\"title\" class=\"title\">Determining Average Velocity from a Graph of Displacement versus Time: Jet Car<\/div>\n<p id=\"import-auto-id2092422\">Find the average velocity of the car whose position is graphed in   <a href=\"#import-auto-id2574769\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<p id=\"import-auto-id4020586\"><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id2362743\">The slope of a graph of <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;\" \/> vs. <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;\" \/> is average velocity, since slope equals rise over run. In this case, rise = change in displacement and run = change in time, so that<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id1850584\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-e9287faf60fba9515b3f6d7c5875d3f5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#108;&#111;&#112;&#101;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#120;&#125;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#116;&#125;&#61;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#45;&#125;&#123;&#118;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"124\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id1666708\">Since the slope is constant here, any two points on the graph can be used to find the slope. (Generally speaking, it is most accurate to use two widely separated points on the straight line. This is because any error in reading data from the graph is proportionally smaller if the interval is larger.)<\/p>\n<p id=\"import-auto-id3539933\"><strong>Solution<\/strong><\/p>\n<p id=\"import-auto-id3577097\">1. Choose two points on the line. In this case, we choose the points labeled on the graph: (6.4 s, 2000 m) and (0.50 s, 525 m). (Note, however, that you could choose any two points.)<\/p>\n<p id=\"import-auto-id1387973\">2. Substitute the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-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;\" \/> values of the chosen points into the equation. Remember in calculating change <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-fc569c1bf69e8005b8f411dddf5640a3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"27\" style=\"vertical-align: -4px;\" \/> we always use final value minus initial value.<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id4181834\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-054c23b567c7b7a4fe56ddd6f20eb2d6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#45;&#125;&#123;&#118;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#120;&#125;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#116;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#48;&#48;&#48;&#32;&#109;&#125;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#50;&#53;&#32;&#109;&#125;&#125;&#123;&#54;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#32;&#115;&#125;&#45;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#48;&#32;&#115;&#125;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"174\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id1731744\">yielding<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2024433\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-095e39b8a8cca8e000875742e60ad393_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#45;&#125;&#123;&#118;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#53;&#48;&#32;&#109;&#47;&#115;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"96\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"import-auto-id2589974\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id4173862\">This is an impressively large land speed (900 km\/h, or about 560 mi\/h): much greater than the typical highway speed limit of 60 mi\/h (27 m\/s or 96 km\/h), but considerably shy of the record of 343 m\/s (1234 km\/h or 766 mi\/h) set in 1997.<\/p>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id2570304\">\n<h1 data-type=\"title\">Graphs of Motion when <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 constant but <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-df2afd8a9cb6add6da009531338a6949_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#92;&#110;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"42\" style=\"vertical-align: -4px;\" \/><\/h1>\n<p id=\"import-auto-id4044849\">The graphs in <a href=\"#import-auto-id3596921\" class=\"autogenerated-content\">(Figure)<\/a> below represent the motion of the jet-powered car as it accelerates toward its top speed, but only during the time when its acceleration is constant. Time starts at zero for this motion (as if measured with a stopwatch), and the displacement and velocity are initially 200 m and 15 m\/s, respectively.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id3596921\">\n<div class=\"bc-figcaption figcaption\">Graphs of motion of a jet-powered car during the time span when its acceleration is constant. (a) The slope of an <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;\" \/> vs. <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;\" \/> graph is velocity. This is shown at two points, and the instantaneous velocities obtained are plotted in the next graph. Instantaneous velocity at any point is the slope of the tangent at that point. (b) The slope of the <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;\" \/> vs. <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;\" \/> graph is constant for this part of the motion, indicating constant acceleration. (c) Acceleration has the constant value of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-74a9ed223ebe63523dcf090c1cc0ff34_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#32;&#109;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"66\" style=\"vertical-align: -4px;\" \/> over the time interval plotted.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id3596922\" data-alt=\"Three line graphs. First is a line graph of displacement over time. Line has a positive slope that increases with time. Second line graph is of velocity over time. Line is straight with a positive slope. Third line graph is of acceleration over time. Line is straight and horizontal, indicating constant acceleration.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_02_07_03.jpg\" data-media-type=\"image\/jpg\" alt=\"Three line graphs. First is a line graph of displacement over time. Line has a positive slope that increases with time. Second line graph is of velocity over time. Line is straight with a positive slope. Third line graph is of acceleration over time. Line is straight and horizontal, indicating constant acceleration.\" width=\"300\" \/><\/span><\/p>\n<\/div>\n<div class=\"bc-figure figure\" id=\"import-auto-id3583460\">\n<div class=\"bc-figcaption figcaption\">A U.S. Air Force jet car speeds down a track. (credit: Matt Trostle, Flickr)<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id3583462\" data-alt=\"\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_02_07_03a.jpg\" data-media-type=\"image\/png\" alt=\"\" width=\"300\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id2379100\">The graph of displacement versus time in <a href=\"#import-auto-id3596921\" class=\"autogenerated-content\">(Figure)<\/a>(a) is a curve rather than a straight line. The slope of the curve becomes steeper as time progresses, showing that the velocity is increasing over time. The slope at any point on a displacement-versus-time graph is the instantaneous velocity at that point. It is found by drawing a straight line tangent to the curve at the point of interest and taking the slope of this straight line. Tangent lines are shown for two points in <a href=\"#import-auto-id3596921\" class=\"autogenerated-content\">(Figure)<\/a>(a). If this is done at every point on the curve and the values are plotted against time, then the graph of velocity versus time shown in <a href=\"#import-auto-id3596921\" class=\"autogenerated-content\">(Figure)<\/a>(b) is obtained. Furthermore, the slope of the graph of velocity versus time is acceleration, which is shown in <a href=\"#import-auto-id3596921\" class=\"autogenerated-content\">(Figure)<\/a>(c).<\/p>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id1516659\">\n<div data-type=\"title\" class=\"title\">Determining Instantaneous Velocity from the Slope at a Point: Jet Car<\/div>\n<p id=\"import-auto-id1752646\">Calculate the velocity of the jet car at a time of 25 s by finding the slope of the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/> vs. <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;\" \/> graph in the graph below.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id4141386\">\n<div class=\"bc-figcaption figcaption\">The slope of an <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;\" \/> vs. <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;\" \/> graph is velocity. This is shown at two points. Instantaneous velocity at any point is the slope of the tangent at that point.\n      <\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1550759\" data-alt=\"A graph of displacement versus time for a jet car. The x axis for time runs from zero to thirty five seconds. The y axis for displacement runs from zero to three thousand meters. The curve depicting displacement is concave up. The slope of the curve increases over time. Slope equals velocity v. There are two points on the curve, labeled, P and Q. P is located at time equals ten seconds. Q is located and time equals twenty-five seconds. A line tangent to P at ten seconds is drawn and has a slope delta x sub P over delta t sub p. A line tangent to Q at twenty five seconds is drawn and has a slope equal to delta x sub q over delta t sub q. Select coordinates are given in a table and consist of the following: time zero seconds displacement two hundred meters; time five seconds displacement three hundred thirty eight meters; time ten seconds displacement six hundred meters; time fifteen seconds displacement nine hundred eighty eight meters. Time twenty seconds displacement one thousand five hundred meters; time twenty five seconds displacement two thousand one hundred thirty eight meters; time thirty seconds displacement two thousand nine hundred meters.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/graphics4-1.jpg\" data-media-type=\"image\/jpg\" alt=\"A graph of displacement versus time for a jet car. The x axis for time runs from zero to thirty five seconds. The y axis for displacement runs from zero to three thousand meters. The curve depicting displacement is concave up. The slope of the curve increases over time. Slope equals velocity v. There are two points on the curve, labeled, P and Q. P is located at time equals ten seconds. Q is located and time equals twenty-five seconds. A line tangent to P at ten seconds is drawn and has a slope delta x sub P over delta t sub p. A line tangent to Q at twenty five seconds is drawn and has a slope equal to delta x sub q over delta t sub q. Select coordinates are given in a table and consist of the following: time zero seconds displacement two hundred meters; time five seconds displacement three hundred thirty eight meters; time ten seconds displacement six hundred meters; time fifteen seconds displacement nine hundred eighty eight meters. Time twenty seconds displacement one thousand five hundred meters; time twenty five seconds displacement two thousand one hundred thirty eight meters; time thirty seconds displacement two thousand nine hundred meters.\" width=\"300\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id4041504\"><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id2333254\">The slope of a curve at a point is equal to the slope of a straight line tangent to the curve at that point. This principle is illustrated in <a href=\"#import-auto-id4141386\" class=\"autogenerated-content\">(Figure)<\/a>, where Q is the point at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c1e1cc568b8dc58d8fce5a88ac7db7ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#53;&#32;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"61\" style=\"vertical-align: 0px;\" \/>.<\/p>\n<p id=\"import-auto-id1729462\"><strong>Solution<\/strong><\/p>\n<p id=\"import-auto-id4063815\">1. Find the tangent line to the curve at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c1e1cc568b8dc58d8fce5a88ac7db7ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#53;&#32;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"61\" style=\"vertical-align: 0px;\" \/>. <\/p>\n<p id=\"import-auto-id3531643\">2. Determine the endpoints of the tangent. These correspond to a position of 1300 m at time 19 s and a position of 3120 m at time 32 s.<\/p>\n<p id=\"import-auto-id2034293\">3. Plug these endpoints into the equation to solve for the slope, <em data-effect=\"italics\"><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;\" \/><\/em>. <\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id3627758\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7332fdfd7dcd7115355a5c3c61150a8c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#108;&#111;&#112;&#101;&#125;&#61;&#123;&#118;&#125;&#95;&#123;&#81;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#120;&#125;&#95;&#123;&#81;&#125;&#125;&#123;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#116;&#125;&#95;&#123;&#81;&#125;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#49;&#50;&#48;&#32;&#109;&#125;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#51;&#48;&#48;&#32;&#109;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#50;&#32;&#115;&#125;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#57;&#32;&#115;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"30\" width=\"274\" style=\"vertical-align: -10px;\" \/><\/div>\n<p id=\"import-auto-id2325601\">Thus,<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id1657105\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-63b493c300a82a90b5ee26ac3fe90474_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#81;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#56;&#50;&#48;&#32;&#109;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#51;&#32;&#115;&#125;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#52;&#48;&#32;&#109;&#47;&#115;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"182\" style=\"vertical-align: -7px;\" \/><\/div>\n<p id=\"import-auto-id2364077\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id4020469\">This is the value given in this figure\u2019s table for <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;\" \/> at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c1e1cc568b8dc58d8fce5a88ac7db7ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#53;&#32;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"61\" style=\"vertical-align: 0px;\" \/>. The value of 140 m\/s for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2554129f3046b93bfffb330501a39a7b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#81;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"20\" style=\"vertical-align: -6px;\" \/> is plotted in <a href=\"#import-auto-id4141386\" class=\"autogenerated-content\">(Figure)<\/a>. The entire graph of <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;\" \/> vs. <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;\" \/> can be obtained in this fashion.<\/p>\n<\/div>\n<p id=\"import-auto-id1690009\">Carrying this one step further, we note that the slope of a velocity versus time graph is acceleration. Slope is rise divided by run; on a <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;\" \/> vs. <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;\" \/> graph, rise = change in velocity <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f415f994bf3cb959366e2e3acbe0a1ee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#118;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"24\" style=\"vertical-align: 0px;\" \/> and run = change in time <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0714636704a254c71bede042781bc57a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"21\" style=\"vertical-align: 0px;\" \/>.<\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id1405001\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">The Slope of <em data-effect=\"italics\">v<\/em> vs. <em data-effect=\"italics\">t<\/em><\/div>\n<p id=\"import-auto-id737611\">The slope of a graph of velocity <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ef71511c70f0e4b25cc6bd69f3bc20c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/> vs. time <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b4e3cbf5d4c5c6d9b702dd139f14c147_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"6\" style=\"vertical-align: 0px;\" \/> is 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;\" \/><em data-effect=\"italics\">.<\/em><\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id4096826\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6fa0b157d98da50637e6272f0b024b2b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#108;&#111;&#112;&#101;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#118;&#125;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#116;&#125;&#61;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"118\" style=\"vertical-align: -6px;\" \/><\/div>\n<\/div>\n<p id=\"import-auto-id4021093\">Since the velocity versus time graph in <a href=\"#import-auto-id3596921\" class=\"autogenerated-content\">(Figure)<\/a>(b) is a straight line, its slope is the same everywhere, implying that acceleration is constant. Acceleration versus time is graphed in <a href=\"#import-auto-id3596921\" class=\"autogenerated-content\">(Figure)<\/a>(c).<\/p>\n<p id=\"import-auto-id4073634\">Additional general information can be obtained from <a href=\"#import-auto-id4141386\" class=\"autogenerated-content\">(Figure)<\/a> and the expression for a straight line, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3da43b1120b85198243b1ee39591d4ed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#120;&#125;&#43;&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"87\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<p>In this case, the vertical axis <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/> is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-63ada879859a9e41fd935f035b7313bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#86;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/>, the intercept <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f56d50c26583f9a035ff6b4e3c0ca5c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"8\" style=\"vertical-align: 0px;\" \/> is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c3b9ce7297f522a77c357066d17856a7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"16\" style=\"vertical-align: -3px;\" \/>, the slope <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6b41df788161942c6f98604d37de8098_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"15\" style=\"vertical-align: 0px;\" \/> is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/>, and the horizontal axis <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/> is <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;\" \/>. Substituting these symbols yields<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id1714581\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-139cc915b920f2a435e06e035d15ae8f_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;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"91\" style=\"vertical-align: -3px;\" \/><\/div>\n<p id=\"import-auto-id1763694\">A general relationship for velocity, acceleration, and time has again been obtained from a graph. Notice that this equation was also derived algebraically from other motion equations in <a href=\"\/contents\/ea2bb23c-4fce-4e9d-a46b-3754125da988@10\">Motion Equations for Constant Acceleration in One Dimension<\/a>.<\/p>\n<p id=\"import-auto-id4171982\">It is not accidental that the same equations are obtained by graphical analysis as by algebraic techniques. In fact, an important way to <em data-effect=\"italics\">discover<\/em> physical relationships is to measure various physical quantities and then make graphs of one quantity against another to see if they are correlated in any way. Correlations imply physical relationships and might be shown by smooth graphs such as those above. From such graphs, mathematical relationships can sometimes be postulated. Further experiments are then performed to determine the validity of the hypothesized relationships.<\/p>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id2306208\">\n<h1 data-type=\"title\">Graphs of Motion Where Acceleration is Not Constant<\/h1>\n<p id=\"import-auto-id1544762\">Now consider the motion of the jet car as it goes from 165 m\/s to its top velocity of 250 m\/s, graphed in <a href=\"#import-auto-id1534076\" class=\"autogenerated-content\">(Figure)<\/a>. Time again starts at zero, and the initial displacement and velocity are 2900 m and 165 m\/s, respectively. (These were the final displacement and velocity of the car in the motion graphed in <a href=\"#import-auto-id3596921\" class=\"autogenerated-content\">(Figure)<\/a>.) Acceleration gradually decreases from <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-74a9ed223ebe63523dcf090c1cc0ff34_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#32;&#109;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"66\" style=\"vertical-align: -4px;\" \/> to zero when the car hits 250 m\/s. The slope of the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/> vs. <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;\" \/> graph increases until <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4046b89cc2a4ca987cb24e18b9193651_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#53;&#32;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"61\" style=\"vertical-align: 0px;\" \/>, after which time the slope is constant. Similarly, velocity increases until 55 s and then becomes constant, since acceleration decreases to zero at 55 s and remains zero afterward.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1534076\">\n<div class=\"bc-figcaption figcaption\">Graphs of motion of a jet-powered car as it reaches its top velocity. This motion begins where the motion in <a href=\"#import-auto-id3596921\" class=\"autogenerated-content\">(Figure)<\/a> ends. (a) The slope of this graph is velocity; it is plotted in the next graph. (b) The velocity gradually approaches its top value. The slope of this graph is acceleration; it is plotted in the final graph. (c) Acceleration gradually declines to zero when velocity becomes constant.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1534078\" data-alt=\"Three line graphs of jet car displacement, velocity, and acceleration, respectively. First line graph is of position over time. Line is straight with a positive slope. Second line graph is of velocity over time. Line graph has a positive slope that decreases over time and flattens out at the end. Third line graph is of acceleration over time. Line has a negative slope that increases over time until it flattens out at the end. The line is not smooth, but has several kinks.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_02_07_04.jpg\" data-media-type=\"image\/jpg\" alt=\"Three line graphs of jet car displacement, velocity, and acceleration, respectively. First line graph is of position over time. Line is straight with a positive slope. Second line graph is of velocity over time. Line graph has a positive slope that decreases over time and flattens out at the end. Third line graph is of acceleration over time. Line has a negative slope that increases over time until it flattens out at the end. The line is not smooth, but has several kinks.\" width=\"350\" \/><\/span><\/p>\n<\/div>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id1406638\">\n<div data-type=\"title\" class=\"title\">Calculating Acceleration from a Graph of Velocity versus Time<\/div>\n<p id=\"import-auto-id1364942\">Calculate the acceleration of the jet car at a time of 25 s by finding the slope of the <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;\" \/> vs. <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;\" \/> graph in <a href=\"#import-auto-id1534076\" class=\"autogenerated-content\">(Figure)<\/a>(b).<\/p>\n<p id=\"import-auto-id3600663\"><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id2165878\">The slope of the curve at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c1e1cc568b8dc58d8fce5a88ac7db7ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#53;&#32;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"61\" style=\"vertical-align: 0px;\" \/> is equal to the slope of the line tangent at that point, as illustrated in <a href=\"#import-auto-id1534076\" class=\"autogenerated-content\">(Figure)<\/a>(b).<\/p>\n<p id=\"import-auto-id945644\"><strong>Solution<\/strong><\/p>\n<p id=\"import-auto-id4081606\">Determine endpoints of the tangent line from the figure, and then plug them into the equation to solve for slope, <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;\" \/>.<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id3503054\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-bb3904a17269ce71704024d9635bac0a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#108;&#111;&#112;&#101;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#118;&#125;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#116;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#54;&#48;&#32;&#109;&#47;&#115;&#125;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#49;&#48;&#32;&#109;&#47;&#115;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#49;&#32;&#115;&#125;&#45;&#49;&#46;&#48;&#32;&#115;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"231\" style=\"vertical-align: -9px;\" \/><\/div>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2028886\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f5825594d74fd42c6f1d61e96ce7d5a3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#48;&#32;&#109;&#47;&#115;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#48;&#32;&#115;&#125;&#125;&#61;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&#32;&#109;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"168\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id1568664\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id2186238\">Note that this 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;\" \/> is consistent with the value plotted in <a href=\"#import-auto-id1534076\" class=\"autogenerated-content\">(Figure)<\/a>(c) at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c1e1cc568b8dc58d8fce5a88ac7db7ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#53;&#32;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"61\" style=\"vertical-align: 0px;\" \/>.<\/p>\n<\/div>\n<p id=\"eip-788\">A graph of displacement versus time can be used to generate a graph of velocity versus time, and a graph of velocity versus time can be used to generate a graph of acceleration versus time. We do this by finding the slope of the graphs at every point. If the graph is linear (i.e., a line with a constant slope), it is easy to find the slope at any point and you have the slope for every point. Graphical analysis of motion can be used to describe both specific and general characteristics of kinematics. Graphs can also be used for other topics in physics. An important aspect of exploring physical relationships is to graph them and look for underlying relationships.<\/p>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1571006\" 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-id1429801\">\n<p id=\"import-auto-id2305974\">A graph of velocity vs. time of a ship coming into a harbor is shown below. (a) Describe the motion of the ship based on the graph. (b)What would a graph of the ship\u2019s acceleration look like? <\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id3504346\"><span data-type=\"media\" id=\"import-auto-id1957571\" data-alt=\"Line graph of velocity versus time. The line has three legs. The first leg is flat. The second leg has a negative slope. The third leg also has a negative slope, but the slope is not as negative as the second leg.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_02_07_04a.jpg\" data-media-type=\"image\/jpg\" alt=\"Line graph of velocity versus time. The line has three legs. The first leg is flat. The second leg has a negative slope. The third leg also has a negative slope, but the slope is not as negative as the second leg.\" width=\"200\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1658952\">\n<p id=\"import-auto-id4086286\">(a) The ship moves at constant velocity and then begins to decelerate at a constant rate. At some point, its deceleration rate decreases. It maintains this lower deceleration rate until it stops moving.<\/p>\n<p id=\"import-auto-id1850777\">(b) A graph of acceleration vs. time would show zero acceleration in the first leg, large and constant negative acceleration in the second leg, and constant negative acceleration.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1666671\"><span data-type=\"media\" id=\"import-auto-id1666672\" data-alt=\"A line graph of acceleration versus time. There are three legs of the graph. All three legs are flat and straight. The first leg shows constant acceleration of 0. The second leg shows a constant negative acceleration. The third leg shows a constant negative acceleration that is not as negative as the second leg.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_02_07_04b.jpg\" data-media-type=\"image\/jpg\" alt=\"A line graph of acceleration versus time. There are three legs of the graph. All three legs are flat and straight. The first leg shows constant acceleration of 0. The second leg shows a constant negative acceleration. The third leg shows a constant negative acceleration that is not as negative as the second leg.\" width=\"200\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"section-summary\" data-depth=\"1\" id=\"fs-id1762928\">\n<h1 data-type=\"title\">Section Summary<\/h1>\n<ul id=\"eip-id4070172\">\n<li id=\"import-auto-id2388505\">Graphs of motion can be used to analyze motion. <\/li>\n<li id=\"import-auto-id4097898\">Graphical solutions yield identical solutions to mathematical methods for deriving motion equations.<\/li>\n<li id=\"import-auto-id2294483\">The slope of a graph of displacement <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;\" \/> vs. time <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b4e3cbf5d4c5c6d9b702dd139f14c147_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"6\" style=\"vertical-align: 0px;\" \/> is velocity <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ef71511c70f0e4b25cc6bd69f3bc20c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/><em data-effect=\"italics\">.<\/em><\/li>\n<li id=\"import-auto-id2025741\">The slope of a graph of velocity <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ef71511c70f0e4b25cc6bd69f3bc20c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/><em data-effect=\"italics\"> vs. time <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b4e3cbf5d4c5c6d9b702dd139f14c147_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"6\" style=\"vertical-align: 0px;\" \/> graph is 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;\" \/><em data-effect=\"italics\">.<\/em><\/em><\/li>\n<li id=\"import-auto-id1561758\">Average velocity, instantaneous velocity, and acceleration can all be obtained by analyzing graphs.<\/li>\n<\/ul>\n<\/div>\n<div class=\"conceptual-questions\" data-depth=\"1\" data-element-type=\"conceptual-questions\">\n<h1 data-type=\"title\">Conceptual Questions<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1550042\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id4182585\">\n<p id=\"import-auto-id3510347\">(a) Explain how you can use the graph of position versus time in <a href=\"#import-auto-id4064025\" class=\"autogenerated-content\">(Figure)<\/a> to describe the change in velocity over time. Identify (b) the time (<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-950c9ae4e34c255ab57a02b935fea600_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#116;&#125;&#95;&#123;&#97;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"13\" style=\"vertical-align: -3px;\" \/>,<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a3fcfa066178d6f12fd037ef82b4be7e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#116;&#125;&#95;&#123;&#98;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"12\" style=\"vertical-align: -3px;\" \/>,<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1b4f3b79278c9340a94daa42bda5bfaa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#116;&#125;&#95;&#123;&#99;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"12\" style=\"vertical-align: -3px;\" \/>,<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-fb12a14330e9bb1ea3aee3bdc221c324_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#116;&#125;&#95;&#123;&#100;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"14\" style=\"vertical-align: -3px;\" \/>, or<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d77e146869dee9523f10a6aba756a97f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#116;&#125;&#95;&#123;&#101;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"12\" style=\"vertical-align: -3px;\" \/>) at which the instantaneous velocity is greatest, (c) the time at which it is zero, and (d) the time at which it is negative.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id4064025\"><span data-type=\"media\" id=\"import-auto-id4033064\" data-alt=\"Line graph of position versus time with 5 points labeled: a, b, c, d, and e. The slope of the line changes. It begins with a positive slope that decreases over time until around point d, where it is flat. It then has a slightly negative slope.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_08Sol_01.jpg\" data-media-type=\"image\/jpg\" alt=\"Line graph of position versus time with 5 points labeled: a, b, c, d, and e. The slope of the line changes. It begins with a positive slope that decreases over time until around point d, where it is flat. It then has a slightly negative slope.\" width=\"300\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id4168594\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1759884\">\n<p id=\"import-auto-id1784242\">(a) Sketch a graph of velocity versus time corresponding to the graph of displacement versus time given in <a href=\"#import-auto-id2562897\" class=\"autogenerated-content\">(Figure)<\/a>. (b) Identify the time or times (<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-950c9ae4e34c255ab57a02b935fea600_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#116;&#125;&#95;&#123;&#97;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"13\" style=\"vertical-align: -3px;\" \/>,<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a3fcfa066178d6f12fd037ef82b4be7e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#116;&#125;&#95;&#123;&#98;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"12\" style=\"vertical-align: -3px;\" \/>,<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1b4f3b79278c9340a94daa42bda5bfaa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#116;&#125;&#95;&#123;&#99;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"12\" style=\"vertical-align: -3px;\" \/>, etc.) at which the instantaneous velocity is greatest. (c) At which times is it zero? (d) At which times is it negative?<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id2562897\"><span data-type=\"media\" id=\"import-auto-id2562898\" data-alt=\"Line graph of position over time with 12 points labeled a through l. Line has a negative slope from a to c, where it turns and has a positive slope till point e. It turns again and has a negative slope till point g. The slope then increases again till l, where it flattens out.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_08Sol_02.jpg\" data-media-type=\"image\/jpg\" alt=\"Line graph of position over time with 12 points labeled a through l. Line has a negative slope from a to c, where it turns and has a positive slope till point e. It turns again and has a negative slope till point g. The slope then increases again till l, where it flattens out.\" width=\"300\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1549493\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1549495\">\n<p id=\"import-auto-id1778974\">(a) Explain how you can determine the acceleration over time from a velocity versus time graph such as the one in <a href=\"#import-auto-id1778975\" class=\"autogenerated-content\">(Figure)<\/a>. (b) Based on the graph, how does acceleration change over time?<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1778975\"><span data-type=\"media\" id=\"import-auto-id4083120\" data-alt=\"Line graph of velocity over time with two points labeled. Point P is at v 1 t 1. Point Q is at v 2 t 2. The line has a positive slope that increases over time.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_08Sol_04.jpg\" data-media-type=\"image\/jpg\" alt=\"Line graph of velocity over time with two points labeled. Point P is at v 1 t 1. Point Q is at v 2 t 2. The line has a positive slope that increases over time.\" width=\"300\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id4131202\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2086598\">\n<p id=\"import-auto-id4033189\">(a) Sketch a graph of acceleration versus time corresponding to the graph of velocity versus time given in <a href=\"#import-auto-id1447833\" class=\"autogenerated-content\">(Figure)<\/a>. (b) Identify the time or times (<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-950c9ae4e34c255ab57a02b935fea600_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#116;&#125;&#95;&#123;&#97;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"13\" style=\"vertical-align: -3px;\" \/>,<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a3fcfa066178d6f12fd037ef82b4be7e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#116;&#125;&#95;&#123;&#98;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"12\" style=\"vertical-align: -3px;\" \/>,<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1b4f3b79278c9340a94daa42bda5bfaa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#116;&#125;&#95;&#123;&#99;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"12\" style=\"vertical-align: -3px;\" \/>, etc.) at which the acceleration is greatest. (c) At which times is it zero? (d) At which times is it negative?<\/p>\n<p id=\"import-auto-id1447832\">\n<div class=\"bc-figure figure\" id=\"import-auto-id1447833\"><span data-type=\"media\" id=\"import-auto-id1447834\" data-alt=\"Line graph of velocity over time with 12 points labeled a through l. The line has a positive slope from a at the origin to d where it slopes downward to e, and then back upward to h. It then slopes back down to point l at v equals 0.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_08Sol_05.jpg\" data-media-type=\"image\/jpg\" alt=\"Line graph of velocity over time with 12 points labeled a through l. The line has a positive slope from a at the origin to d where it slopes downward to e, and then back upward to h. It then slopes back down to point l at v equals 0.\" width=\"300\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1365827\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id4021330\">\n<p id=\"import-auto-id4047633\">Consider the velocity vs. time graph of a person in an elevator shown in <a href=\"#import-auto-id2006890\" class=\"autogenerated-content\">(Figure)<\/a>. Suppose the elevator is initially at rest. It then accelerates for 3 seconds, maintains that velocity for 15 seconds, then decelerates for 5 seconds until it stops. The acceleration for the entire trip is not constant so we cannot use the equations of motion from <a href=\"\/contents\/ea2bb23c-4fce-4e9d-a46b-3754125da988@10\">Motion Equations for Constant Acceleration in One Dimension<\/a> for the complete trip. (We could, however, use them in the three individual sections where acceleration is a constant.) Sketch graphs of (a) position vs. time and (b) acceleration vs. time for this trip.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id2006890\"><span data-type=\"media\" id=\"import-auto-id2006892\" data-alt=\"Line graph of velocity versus time. Line begins at the origin and has a positive slope until it reaches 3 meters per second at 3 seconds. The slope is then zero until 18 seconds, where it becomes negative until the line reaches a velocity of 0 at 23 seconds.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_03_08Sol_07.jpg\" data-media-type=\"image\/jpg\" alt=\"Line graph of velocity versus time. Line begins at the origin and has a positive slope until it reaches 3 meters per second at 3 seconds. The slope is then zero until 18 seconds, where it becomes negative until the line reaches a velocity of 0 at 23 seconds.\" width=\"350\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2576953\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2589937\">\n<p id=\"import-auto-id4124847\">A cylinder is given a push and then rolls up an inclined plane. If the origin is the starting point, sketch the position, velocity, and acceleration of the cylinder vs. time as it goes up and then down the plane. <\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"problems-exercises\" data-depth=\"1\" id=\"fs-id1987308\" data-element-type=\"problems-exercises\">\n<h1 data-type=\"title\">Problems &amp; Exercises<\/h1>\n<p id=\"import-auto-id1960253\">Note: There is always uncertainty in numbers taken from graphs. If your answers differ from expected values, examine them to see if they are within data extraction uncertainties estimated by you.<\/p>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id4088406\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id4088408\">\n<p id=\"import-auto-id1729859\">(a) By taking the slope of the curve in <a href=\"#import-auto-id1798398\" class=\"autogenerated-content\">(Figure)<\/a>, verify that the velocity of the jet car is 115 m\/s at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-548846017f52d3443ff973ff19b3faeb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#48;&#32;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"61\" style=\"vertical-align: 0px;\" \/>. (b) By taking the slope of the curve at any point in <a href=\"#import-auto-id4101417\" class=\"autogenerated-content\">(Figure)<\/a>, verify that the jet car\u2019s acceleration is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-74a9ed223ebe63523dcf090c1cc0ff34_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#32;&#109;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"66\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1798398\"><span data-type=\"media\" id=\"import-auto-id4028860\" data-alt=\"Line graph of position over time. Line has positive slope that increases over time.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_02_08Sol_11.jpg\" data-media-type=\"image\/jpg\" alt=\"Line graph of position over time. Line has positive slope that increases over time.\" width=\"350\" \/><\/span><\/div>\n<div class=\"bc-figure figure\" id=\"import-auto-id4101417\"><span data-type=\"media\" id=\"import-auto-id4101418\" data-alt=\"Line graph of velocity versus time. Line is straight with a positive slope.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_02_08Sol_12.jpg\" data-media-type=\"image\/jpg\" alt=\"Line graph of velocity versus time. Line is straight with a positive slope.\" width=\"350\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2295253\">\n<p id=\"import-auto-id952467\">(a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f011dfa2b72f8f6ae713929c3b9f56d7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#49;&#53;&#32;&#109;&#47;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"63\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"import-auto-id1510856\">(b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-74a9ed223ebe63523dcf090c1cc0ff34_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#32;&#109;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"66\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id4012994\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id4012996\">\n<p id=\"import-auto-id2006034\">Using approximate values, calculate the slope of the curve in <a href=\"#import-auto-id4122996\" class=\"autogenerated-content\">(Figure)<\/a> to verify that the velocity at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-01fc0300cba41dffb3bd4e3f474cfd41_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#46;&#48;&#32;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"74\" style=\"vertical-align: -1px;\" \/> is 0.208 m\/s. Assume all values are known to 3 significant figures.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id4122996\"><span data-type=\"media\" id=\"import-auto-id2015268\" data-alt=\"Line graph of position versus time. Line is straight with a positive slope.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_02_08Sol_13.jpg\" data-media-type=\"image\/jpg\" alt=\"Line graph of position versus time. Line is straight with a positive slope.\" width=\"350\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1770908\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1770911\">\n<p id=\"import-auto-id1761648\">Using approximate values, calculate the slope of the curve in <a href=\"#import-auto-id4122996\" class=\"autogenerated-content\">(Figure)<\/a> to verify that the velocity at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-e8b2f7c83e5c33bbe0b9f89d3134cc3e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#48;&#46;&#48;&#32;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"74\" style=\"vertical-align: 0px;\" \/> is 0.238 m\/s. Assume all values are known to 3 significant figures.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1707522\">\n<div data-type=\"equation\" class=\"equation\" id=\"eip-id2453534\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ee00d6de1011e6ed6153a006f6c2ef70_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#49;&#46;&#55;&#125;&#45;&#54;&#46;&#57;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#51;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#32;&#45;&#32;&#50;&#48;&#125;&#46;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#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;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#51;&#56;&#32;&#109;&#47;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"231\" style=\"vertical-align: -9px;\" \/><\/div>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2475925\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1744756\">\n<p id=\"import-auto-id1743717\">By taking the slope of the curve in <a href=\"#import-auto-id3552017\" class=\"autogenerated-content\">(Figure)<\/a>, verify that the acceleration is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4ccb090e987a24618192f5f536c7ccf4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#50;&#32;&#109;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"61\" style=\"vertical-align: -4px;\" \/> at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-95e655048df653a584e0216bfd727ed8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#32;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"61\" style=\"vertical-align: -1px;\" \/>.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id3552017\"><span data-type=\"media\" id=\"import-auto-id3552018\" data-alt=\"Line graph of velocity versus time. Line has a positive slope that decreases over time until the line flattens out.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_02_08Sol_14.jpg\" data-media-type=\"image\/jpg\" alt=\"Line graph of velocity versus time. Line has a positive slope that decreases over time until the line flattens out.\" width=\"350\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1372323\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1544965\">\n<p id=\"import-auto-id3575253\">Construct the displacement graph for the subway shuttle train as shown in <a href=\"\/contents\/6023b87d-5a28-4910-9e51-ee7fd11c98e1@4#import-auto-id2590556\" class=\"autogenerated-content\">(Figure)<\/a>(a). Your graph should show the position of the train, in kilometers, from t = 0 to 20 s. You will need to use the information on acceleration and velocity given in the examples for this figure. <\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1778988\">\n<div class=\"bc-figure figure\" id=\"import-auto-id3597350\"><span data-type=\"media\" id=\"import-auto-id3597352\" data-alt=\"Line graph of position versus time. Line begins with a slight positive slope. It then kinks to a much greater positive slope.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_02_08Sol_15.jpg\" data-media-type=\"image\/jpg\" alt=\"Line graph of position versus time. Line begins with a slight positive slope. It then kinks to a much greater positive slope.\" width=\"350\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2290187\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2290189\">\n<p id=\"import-auto-id2592264\">(a) Take the slope of the curve in <a href=\"#import-auto-id4064858\" class=\"autogenerated-content\">(Figure)<\/a> to find the jogger\u2019s velocity at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3e14c50c6dcaab1b5dd03d216af26992_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#61;&#50;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#53;&#32;&#115;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"61\" style=\"vertical-align: 0px;\" \/>. (b) Repeat at 7.5 s. These values must be consistent with the graph in <a href=\"#import-auto-id4128350\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id4064858\"><span data-type=\"media\" id=\"import-auto-id4064859\" data-alt=\"Line graph of position over time. Line begins sloping upward, then kinks back down, then kinks back upward again.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_02_08Sol_16.jpg\" data-media-type=\"image\/jpg\" alt=\"Line graph of position over time. Line begins sloping upward, then kinks back down, then kinks back upward again.\" width=\"300\" \/><\/span><\/div>\n<div class=\"bc-figure figure\" id=\"import-auto-id4128350\"><span data-type=\"media\" id=\"import-auto-id4128351\" data-alt=\"Line graph of velocity over time. Line begins with a positive slope, then kinks downward with a negative slope, then kinks back upward again. It kinks back down again slightly, then back up again, and ends with a slightly less positive slope.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_02_08Sol_17.jpg\" data-media-type=\"image\/jpg\" alt=\"Line graph of velocity over time. Line begins with a positive slope, then kinks downward with a negative slope, then kinks back upward again. It kinks back down again slightly, then back up again, and ends with a slightly less positive slope.\" width=\"300\" \/><\/span><\/div>\n<div class=\"bc-figure figure\" id=\"import-auto-id4151339\"><span data-type=\"media\" id=\"import-auto-id4151340\" data-alt=\"\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_02_08Sol_18.jpg\" data-media-type=\"image\/jpg\" alt=\"\" width=\"300\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3520768\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1434602\">\n<p id=\"import-auto-id1820030\">A graph of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5dbd503d889d025c31c055b7b26509b3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;&#92;&#108;&#101;&#102;&#116;&#40;&#116;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"31\" style=\"vertical-align: -4px;\" \/> is shown for a world-class track sprinter in a 100-m race. (See  <a href=\"#import-auto-id4125036\" class=\"autogenerated-content\">(Figure)<\/a>). (a) What is his average velocity for the first 4 s? (b) What is his instantaneous velocity at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b0ae02577c0382a3e9644bb27ce4c72c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#61;&#53;&#32;&#115;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"47\" style=\"vertical-align: 0px;\" \/>? (c) What is his average acceleration between 0 and 4 s? (d) What is his time for the race?<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id4125036\"><span data-type=\"media\" id=\"import-auto-id4125037\" data-alt=\"Line graph of velocity versus time. The line has two legs. The first has a constant positive slope. The second is flat, with a slope of 0.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_02_08Sol_20.jpg\" data-media-type=\"image\/jpg\" alt=\"Line graph of velocity versus time. The line has two legs. The first has a constant positive slope. The second is flat, with a slope of 0.\" width=\"350\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1778256\">\n<p id=\"import-auto-id1746146\">(a) 6 m\/s<\/p>\n<p id=\"import-auto-id1746150\">(b) 12 m\/s<\/p>\n<p id=\"import-auto-id3624887\">(c) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-dc0e5863e5c980ac7ccdeb6dc64d192a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#32;&#109;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"52\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"import-auto-id1757765\">(d) 10 s<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1582774\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1598940\">\n<p id=\"import-auto-id1730106\"><a href=\"#import-auto-id4035681\" class=\"autogenerated-content\">(Figure)<\/a> shows the displacement graph for a particle for 5 s. Draw the corresponding velocity and acceleration graphs. <\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id4035681\"><span data-type=\"media\" id=\"import-auto-id4035682\" data-alt=\"Line graph of position versus time. The line has 4 legs. The first leg has a positive slope. The second leg has a negative slope. The third has a slope of 0. The fourth has a positive slope.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_02_08Sol_21.jpg\" data-media-type=\"image\/jpg\" alt=\"Line graph of position versus time. The line has 4 legs. The first leg has a positive slope. The second leg has a negative slope. The third has a slope of 0. The fourth has a positive slope.\" width=\"350\" \/><\/span><\/div>\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=\"fs-id1218535\">\n<dt>independent variable<\/dt>\n<dd id=\"fs-id1343248\">the variable that the dependent variable is measured with respect to; usually plotted along the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/>-axis\n<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id1845390\">\n<dt>dependent variable<\/dt>\n<dd id=\"fs-id4015576\">the variable that is being measured; usually plotted along the <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;\" \/>-axis\n<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id3600469\">\n<dt>slope<\/dt>\n<dd id=\"fs-id1544972\">the difference in <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;\" \/>-value (the rise) divided by the difference in <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;\" \/>-value (the run) of two points on a straight line<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id4021637\">\n<dt>y-intercept<\/dt>\n<dd id=\"fs-id3525350\">the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a609f505a979b0cc9ff9ab74cceca58e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: -4px;\" \/>value when <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;\" \/><em data-effect=\"italics\">= 0, or when the graph crosses the <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;\" \/>-axis<\/em><\/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-143","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\/143","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\/143\/revisions"}],"predecessor-version":[{"id":144,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/143\/revisions\/144"}],"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\/143\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/media?parent=143"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapter-type?post=143"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/contributor?post=143"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/license?post=143"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}