{"id":807,"date":"2019-08-29T10:33:46","date_gmt":"2019-08-29T14:33:46","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/physicsforlifesciences1phys1108\/?post_type=chapter&#038;p=807"},"modified":"2019-09-16T01:28:46","modified_gmt":"2019-09-16T05:28:46","slug":"2-9-new-motion-with-a-non-constant-acceleration-derivatives","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/physicsforlifesciences1phys1108\/chapter\/2-9-new-motion-with-a-non-constant-acceleration-derivatives\/","title":{"raw":"2.9 New - Motion with a non-constant acceleration - Derivatives","rendered":"2.9 New &#8211; Motion with a non-constant acceleration &#8211; Derivatives"},"content":{"raw":"<div class=\"textbox textbox--learning-objectives\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Learning Objectives<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<ul>\r\n \t<li><span>By the end of this section, you will be able to:<\/span>\r\n<ul>\r\n \t<li>Explain the difference between average velocity and instantaneous velocity.<\/li>\r\n \t<li>Describe the difference between velocity and speed.<\/li>\r\n \t<li>Calculate the instantaneous velocity given the mathematical equation for the velocity.<\/li>\r\n \t<li>Calculate the speed given the instantaneous velocity.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n&nbsp;\r\n\r\n<span>We have now seen how to calculate the average velocity between two positions. However, since objects in the real world move continuously through space and time, we would like to find the velocity of an object at any single point. We can find the velocity of the object anywhere along its path by using some fundamental principles of calculus. This section gives us better insight into the physics of motion and will be useful in later chapters.<\/span>\r\n\r\n&nbsp;\r\n\r\n<a class=\"elm-skip-link\" href=\"https:\/\/phys.libretexts.org\/Bookshelves\/University_Physics\/Book%3A_University_Physics_(OpenStax)\/Map%3A_University_Physics_I_-_Mechanics%2C_Sound%2C_Oscillations%2C_and_Waves_(OpenStax)\/3%3A_Motion_Along_a_Straight_Line\/3.2%3A_Instantaneous_Velocity_and_Speed#elm-main-content\" title=\"Press enter to skip to the main content\">Read this online for free - and legally at Libre Texts . Click here<\/a>\r\n\r\n<article id=\"elm-main-content\" class=\"elm-content-container\"><section class=\"mt-content-container\">\r\n<div class=\"mt-section\" id=\"section_1\">\r\n<h1 class=\"editable\">Instantaneous Velocity<\/h1>\r\nThe quantity that tells us how fast an object is moving anywhere along its path is the<strong>instantaneous velocity<\/strong>, usually called simply<em>velocity<\/em>. It is the average velocity between two points on the path in the limit that the time (and therefore the displacement) between the two points approaches zero. To illustrate this idea mathematically, we need to express position x as a continuous function of t denoted by x(t). The expression for the average velocity between two points using this notation is\r\n\r\n&nbsp;\r\n<div class=\"MathJax_Display\"><\/div>\r\n<div class=\"note1\">\r\n<p class=\"boxtitle\">Instantaneous Velocity<\/p>\r\nThe instantaneous velocity of an object is the limit of the average velocity as the elapsed time approaches zero, or the derivative of x with respect to t:\r\n<div class=\"MathJax_Display\"><\/div>\r\n<\/div>\r\nLike average velocity, instantaneous velocity is a vector with dimension of length per time. The instantaneous velocity at a specific time point t<sub>0<\/sub>is the rate of change of the position function, which is the slope of the position function x(t) at t<sub>0<\/sub>. Figure 3.6 shows how the average velocity\"&gt;<span class=\"math\" id=\"MathJax-Span-121\"><span><span class=\"mrow\" id=\"MathJax-Span-122\"><span class=\"texatom\" id=\"MathJax-Span-123\"><span class=\"mrow\" id=\"MathJax-Span-124\"><span class=\"munderover\" id=\"MathJax-Span-125\"><span class=\"mi\" id=\"MathJax-Span-126\">v<\/span><span class=\"mo\" id=\"MathJax-Span-127\">\u00af<\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-128\">=<\/span><span class=\"mfrac\" id=\"MathJax-Span-129\"><span class=\"mrow\" id=\"MathJax-Span-130\"><span class=\"mi\" id=\"MathJax-Span-131\">\u0394<\/span><span class=\"mi\" id=\"MathJax-Span-132\">x<\/span><\/span><span class=\"mrow\" id=\"MathJax-Span-133\"><span class=\"mi\" id=\"MathJax-Span-134\">\u0394<\/span><span class=\"mi\" id=\"MathJax-Span-135\">t<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">v\u00af=\u0394x\u0394t<\/span>between two times approaches the instantaneous velocity at t<sub>0<\/sub>. The instantaneous velocity is shown at time t<sub>0<\/sub>, which happens to be at the maximum of the position function. The slope of the position graph is zero at this point, and thus the instantaneous velocity is zero. At other times, t<sub>1<\/sub>, t<sub>2<\/sub>, and so on, the instantaneous velocity is not zero because the slope of the position graph would be positive or negative. If the position function had a minimum, the slope of the position graph would also be zero, giving an instantaneous velocity of zero there as well. Thus, the zeros of the velocity function give the minimum and maximum of the position function.\r\n<figure><img alt=\"Graph shows position plotted versus time. Position increases from t1 to t2 and reaches maximum at t0. It decreases to at and continues to decrease at t4. The slope of the tangent line at t0 is indicated as the instantaneous velocity.\" class=\"internal default\" width=\"290px\" height=\"233px\" src=\"https:\/\/phys.libretexts.org\/@api\/deki\/files\/3708\/clipboard_efc61ece1c38ff1f7239a6cabb1619e94?revision=1&amp;size=bestfit&amp;width=290&amp;height=233\" \/><figcaption>Figure<em>In a graph of position versus time, the instantaneous velocity is the slope of the tangent line at a given point. The average velocities.\u00a0<span class=\"MJX_Assistive_MathML\" role=\"presentation\">\u0394<\/span>t \u2192 0, the average velocity approaches the instantaneous velocity at t = t<sub>0<\/sub>.<\/em><\/figcaption><\/figure>\r\n<div class=\"example\">\r\n<p class=\"boxtitle\">Example 3.2: Finding Velocity from a Position-Versus-Time Graph<\/p>\r\nGiven the position-versus-time graph of Figure 3.7, find the velocity-versus-time graph.\r\n<figure><img alt=\"Graph shows position in kilometers plotted as a function of time at minutes. It starts at the origin, reaches 0.5 kilometers at 0.5 minutes, remains constant between 0.5 and 0.9 minutes, and decreases to 0 at 2.0 minutes.\" class=\"internal default\" width=\"307px\" height=\"274px\" src=\"https:\/\/phys.libretexts.org\/@api\/deki\/files\/3709\/clipboard_efaca94677e552a40cfa727e5283ce0bb?revision=1&amp;size=bestfit&amp;width=307&amp;height=274\" \/><figcaption><em>The object starts out in the positive direction, stops for a short time, and then reverses direction, heading back toward the origin. Notice that the object comes to rest instantaneously, which would require an infinite force. Thus, the graph is an approximation of motion in the real world. (The concept of force is discussed in Newton\u2019s Laws of Motion.)<\/em><\/figcaption><\/figure>\r\n<div class=\"mt-section\" id=\"section_2\">\r\n\r\n&nbsp;\r\n<h3 class=\"editable\">Strategy<\/h3>\r\nThe graph contains three straight lines during three time intervals. We find the velocity during each time interval by taking the slope of the line using the grid.\r\n\r\n<\/div>\r\n<div class=\"mt-section\" id=\"section_3\">\r\n\r\n&nbsp;\r\n<h3 class=\"editable\">Solution<\/h3>\r\nThe graph of these values of velocity versus time is shown in Figure 3.8.\r\n<figure><img alt=\"Graph shows velocity in meters per second plotted as a function of time at seconds. The velocity is 1 meter per second between 0 and 0.5 seconds, zero between 0.5 and 1.0 seconds, and -0.5 between 1.0 and 2.0 seconds.\" class=\"internal default\" width=\"298px\" height=\"212px\" src=\"https:\/\/phys.libretexts.org\/@api\/deki\/files\/3710\/clipboard_e7ce331edf68d0702a995c9fe676d5969?revision=1&amp;size=bestfit&amp;width=298&amp;height=212\" \/><figcaption>Figure<em>The velocity is positive for the first part of the trip, zero when the object is stopped, and negative when the object reverses direction.<\/em><\/figcaption><\/figure>\r\n<\/div>\r\n<div class=\"mt-section\" id=\"section_4\">\r\n\r\n&nbsp;\r\n<h3 class=\"editable\">Significance<\/h3>\r\nDuring the time interval between 0 s and 0.5 s, the object\u2019s position is moving away from the origin and the position-versus-time curve has a positive slope. At any point along the curve during this time interval, we can find the instantaneous velocity by taking its slope, which is +1 m\/s, as shown in Figure 3.8. In the subsequent time interval, between 0.5 s and 1.0 s, the position doesn\u2019t change and we see the slope is zero. From 1.0 s to 2.0 s, the object is moving back toward the origin and the slope is \u22120.5 m\/s. The object has reversed direction and has a negative velocity.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"mt-section\" id=\"section_5\">\r\n\r\n&nbsp;\r\n<h1 class=\"editable\">Speed<\/h1>\r\nIn everyday language, most people use the terms speed and velocity interchangeably. In physics, however, they do not have the same meaning and are distinct concepts. One major difference is that speed has no direction; that is, speed is a scalar.\r\n\r\nWe can calculate the\u00a0<strong>average speed\u00a0<\/strong>by finding the total distance traveled divided by the elapsed time:\r\n\r\n<\/div>\r\n<\/section><strong>Table 3.1 Speeds of various objects in metres per second<\/strong>\r\n\r\n<\/article>\u00a0 \u00a0 \u00a0 \u00a0 Continental drift = 10 <sup>-7<\/sup> m\/s\r\n\r\nBrisk walk = 1.7 m\/s\r\n\r\nCyclist = 4. 4 m\/s\r\n\r\nSprint runner = 12.2 m\/s\r\n\r\nOfficial land speed record = 341.4 m\/s\r\n\r\nSpeed of sound = 343 m\/s\r\n\r\nEscape velocity =\u00a0Escape velocity is the velocity at which an object must be launched so that it overcomes Earth\u2019s gravity and is not pulled back toward Earth = 11,200 m\/s\r\n\r\nISS = International Space Station =. 7,600 \u00a0m\/s\r\n\r\nOrbital speed of the Earth =.\u00a029,700 m\/s\r\n\r\nSpeed of light in.a vacuum = 3.00 x 10<sup>8<\/sup> m\/s\r\n\r\n&nbsp;\r\n\r\n<article id=\"elm-main-content\" class=\"elm-content-container\"><section class=\"mt-content-container\">\r\n<div class=\"mt-section\" id=\"section_7\">\r\n<h1 class=\"editable\">Calculating Instantaneous Velocity<\/h1>\r\nWhen calculating instantaneous velocity, we need to specify the explicit form of the position function x(t). For the moment, let\u2019s use polynomials x(t) = At<sup>n<\/sup>, because they are easily differentiated using the power rule of calculus:\r\n<div class=\"MathJax_Display\"><\/div>\r\nThe following example illustrates the use of Equation 3.7.\r\n<div class=\"example\">\r\n<p class=\"boxtitle\">Example 3.3: Instantaneous Velocity Versus Average Velocity<\/p>\r\nThe position of a particle is given by x(t) = 3.0t + 0.5t<sup>3<\/sup>m.\r\n<ol start=\"1\">\r\n \t<li>Using Equation 3.4 and Equation 3.7, find the instantaneous velocity at t = 2.0 s.<\/li>\r\n \t<li>Calculate the average velocity between 1.0 s and 3.0 s.<\/li>\r\n<\/ol>\r\n<div class=\"mt-section\" id=\"section_8\">\r\n\r\n&nbsp;\r\n<h3 class=\"editable\">Strategy<\/h3>\r\nEquation 3.4 give the instantaneous velocity of the particle as the derivative of the position function. Looking at the form of the position function given, we see that it is a polynomial in t. Therefore, we can use Equation 3.7, the power rule from calculus, to find the solution. We use Equation 3.6 to calculate the average velocity of the particle.\r\n\r\n<\/div>\r\n<div class=\"mt-section\" id=\"section_9\">\r\n\r\n[latexpage]\r\n<h3 class=\"editable\">Solution<\/h3>\r\n<ol start=\"1\">\r\n \t<li><span class=\"math\" id=\"MathJax-Span-457\"><span><span class=\"mrow\" id=\"MathJax-Span-458\"><span class=\"mfrac\" id=\"MathJax-Span-459\"><span class=\"mrow\" id=\"MathJax-Span-460\"><span class=\"mi\" id=\"MathJax-Span-461\">\u00a0$$ \\frac {dx}{dt}=v(t)<\/span><\/span><\/span><\/span><\/span><\/span>\u00a0 \/frac $$<\/li>\r\n \t<li>\u00a0dx\/dt = v(t) = 3.0 + 1.5t<sup>2<\/sup>\u00a0<sup>\u00a0<\/sup>m\/s<\/li>\r\n \t<li>Substituting t = 2.0 s into this equation gives v<sub>(2.0 s)<\/sub> = [3.0 + 1.5(2.0)<sup>2<\/sup>] m\/s = 9.0 m\/s.<\/li>\r\n \t<li>To determine the average velocity of the particle between 1.0 s and 3.0 s, we calculate the values of x(1.0 s) and x(3.0 s):<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div class=\"example\">\r\n\r\n&nbsp;\r\n<p class=\"boxtitle\">Example 3.4: Instantaneous Velocity Versus Speed<\/p>\r\nConsider the motion of a particle in which the position is x(t) = 3.0t \u2212 3t<sup>2<\/sup>m.\r\n<ol start=\"1\">\r\n \t<li>What is the instantaneous velocity at t = 0.25 s, t = 0.50 s, and t = 1.0 s?<\/li>\r\n \t<li>What is the speed of the particle at these times?<\/li>\r\n<\/ol>\r\n<div class=\"mt-section\" id=\"section_11\">\r\n\r\n&nbsp;\r\n<h3 class=\"editable\">Strategy<\/h3>\r\nThe instantaneous velocity is the derivative of the position function and the speed is the magnitude of the instantaneous velocity. We use Equation 3.4 and Equation 3.7 to solve for instantaneous velocity.\r\n\r\n<\/div>\r\n<div class=\"mt-section\" id=\"section_12\">\r\n\r\n&nbsp;\r\n<h3 class=\"editable\">Solution<\/h3>\r\n<ol start=\"1\">\r\n \t<li><\/li>\r\n \t<li>v(0.25 s) = 1.50 m\/s, v(0.5 s) = 0 m\/s, v(1.0 s) = \u22123.0 m\/s<\/li>\r\n \t<li>Speed = |v(t)| = 1.50 m\/s, 0.0 m\/s, and 3.0 m\/s<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"mt-section\" id=\"section_13\">\r\n\r\n&nbsp;\r\n<h3 class=\"editable\">Significance<\/h3>\r\nThe velocity of the particle gives us direction information, indicating the particle is moving to the left (west) or right (east). The speed gives the magnitude of the velocity. By graphing the position, velocity, and speed as functions of time, we can understand these concepts visually Figure 3.9. In (a), the graph shows the particle moving in the positive direction until t = 0.5 s, when it reverses direction. The reversal of direction can also be seen in (b) at 0.5 s where the velocity is zero and then turns negative. At 1.0 s it is back at the origin where it started. The particle\u2019s velocity at 1.0 s in (b) is negative, because it is traveling in the negative direction. But in (c), however, its speed is positive and remains positive throughout the travel time. We can also interpret velocity as the slope of the position-versus-time graph. The slope of x(t) is decreasing toward zero, becoming zero at 0.5 s and increasingly negative thereafter. This analysis of comparing the graphs of position, velocity, and speed helps catch errors in calculations. The graphs must be consistent with each other and help interpret the calculations.\r\n<figure><img alt=\"Graph A shows position in meters plotted versus time in seconds. It starts at the origin, reaches maximum at 0.5 seconds, and then start to decrease crossing x axis at 1 second. Graph B shows velocity in meters per second plotted as a function of time at seconds. Velocity linearly decreases from the left to the right. Graph C shows absolute velocity in meters per second plotted as a function of time at seconds. Graph has a V-leeter shape. Velocity decreases till 0.5 seconds; then it starts to increase.\" class=\"internal default\" width=\"596px\" height=\"182px\" src=\"https:\/\/phys.libretexts.org\/@api\/deki\/files\/3732\/clipboard_eba5e0c299a72d21ee1326256584794aa?revision=1&amp;size=bestfit&amp;width=596&amp;height=182\" \/><figcaption>Figure<em>(a) Position: x(t) versus time. (b) Velocity: v(t) versus time. The slope of the position graph is the velocity. A rough comparison of the slopes of the tangent lines in (a) at 0.25 s, 0.5 s, and 1.0 s with the values for velocity at the corresponding times indicates they are the same values. (c) Speed: |v(t)| versus time. Speed is always a positive number.<\/em><\/figcaption><\/figure>\r\n<\/div>\r\n<\/div>\r\n<div class=\"exercise\">\r\n<p class=\"boxtitle\">Check Your Understanding 3.2<\/p>\r\nThe position of an object as a function of time is x(t) = \u22123t<sup>2<\/sup>m. (a) What is the velocity of the object as a function of time? (b) Is the velocity ever positive? (c) What are the velocity and speed at t = 1.0 s?\r\n\r\n<\/div>\r\n<div>\r\n<div class=\"mt-section\" id=\"section_14\">\r\n\r\n&nbsp;\r\n<h1 class=\"editable\">Contributors<\/h1>\r\n<\/div>\r\n<\/div>\r\n<ul>\r\n \t<li><section id=\"main-content\" data-is-baked=\"true\">\r\n<p id=\"eip-677\">Samuel J. Ling (Truman State University),\u00a0Jeff Sanny (Loyola Marymount University), and Bill Moebs\u00a0with many contributing authors. This work is licensed by OpenStax University Physics under a\u00a0<a href=\"http:\/\/creativecommons.org\/licenses\/by\/4.0\/\" rel=\"external nofollow noopener\" target=\"_blank\" class=\"external\">Creative Commons Attribution License (by 4.0)<\/a>.<\/p>\r\n\r\n<\/section>\r\n<h1>Exercises<\/h1>\r\n<h2 class=\"editable\"><a title=\"3.2: Instantaneous Velocity and Speed\" href=\"https:\/\/phys.libretexts.org\/Bookshelves\/University_Physics\/Book%3A_University_Physics_(OpenStax)\/Map%3A_University_Physics_I_-_Mechanics%2C_Sound%2C_Oscillations%2C_and_Waves_(OpenStax)\/3%3A_Motion_Along_a_Straight_Line\/3.2%3A_Instantaneous_Velocity_and_Speed\" rel=\"internal\">Instantaneous Velocity and Speed<\/a><\/h2>\r\n<ol class=\"mt-indent-1\" start=\"30\">\r\n \t<li>A woodchuck runs 20 m to the right in 5 s, then turns and runs 10 m to the left in 3 s. (a) What is the average velocity of the woodchuck? (b) What is its average speed?<\/li>\r\n \t<li>Sketch the velocity-versus-time graph from the following position-versus-time graph.<\/li>\r\n<\/ol>\r\n<img alt=\"Graph shows position in meters plotted versus time in seconds. It starts at the origin, reaches 4 meters at 0.4 seconds; decreases to -2 meters at 0.6 seconds, reaches minimum of -6 meters at 1 second, increases to -4 meters at 1.6 seconds, and reaches 2 meters at 2 seconds.\" class=\"internal default\" width=\"299px\" height=\"276px\" src=\"https:\/\/phys.libretexts.org\/@api\/deki\/files\/3764\/clipboard_e9067e449d507cb185bcd2db18f0f8378?revision=1&amp;size=bestfit&amp;width=299&amp;height=276\" \/>\r\n<ol class=\"mt-indent-1\" start=\"32\">\r\n \t<li>Sketch the velocity-versus-time graph from the following position-versus-time graph.<\/li>\r\n<\/ol>\r\n<img alt=\"Graph shows position plotted versus time in seconds. Graph has a sinusoidal shape. It starts with the positive value at zero time, changes to negative, and then starts to increase.\" class=\"internal default\" width=\"287px\" height=\"204px\" src=\"https:\/\/phys.libretexts.org\/@api\/deki\/files\/3765\/clipboard_e8f2d6f0c158a7dbb0a6473461e708e4a?revision=1&amp;size=bestfit&amp;width=287&amp;height=204\" \/>\r\n<ol class=\"mt-indent-1\" start=\"33\">\r\n \t<li>Given the following velocity-versus-time graph, sketch the position-versus-time graph.<\/li>\r\n<\/ol>\r\n<img alt=\"Graph shows velocity plotted versus time. It starts with the positive value at zero time, decreases to the negative value and remains constant.\" class=\"internal default\" width=\"291px\" height=\"215px\" src=\"https:\/\/phys.libretexts.org\/@api\/deki\/files\/3766\/clipboard_eff678ded18eed80b84f692f0c227a6cc?revision=1&amp;size=bestfit&amp;width=291&amp;height=215\" \/>\r\n<ol class=\"mt-indent-1\" start=\"34\">\r\n \t<li>An object has a position function x(t) = 5t m. (a) What is the velocity as a function of time? (b) Graph the position function and the velocity function. 35. A particle moves along the x-axis according to x(t) = 10t \u2212 2t<sup>2<\/sup> m. (a) What is the instantaneous velocity at t = 2 s and t = 3 s? (b) What is the instantaneous speed at these times? (c) What is the average velocity between t = 2 s and t = 3 s?<\/li>\r\n \t<li><strong>Unreasonable results<\/strong>. A particle moves along the x-axis according to x(t) = 3t<sup>3<\/sup> + 5t . At what time is the velocity of the particle equal to zero? Is this reasonable?<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>\r\n<h2 class=\"editable\"><a title=\"3.6: Finding Velocity and Displacement from Acceleration\" href=\"https:\/\/phys.libretexts.org\/Bookshelves\/University_Physics\/Book%3A_University_Physics_(OpenStax)\/Map%3A_University_Physics_I_-_Mechanics%2C_Sound%2C_Oscillations%2C_and_Waves_(OpenStax)\/3%3A_Motion_Along_a_Straight_Line\/3.6%3A_Finding_Velocity_and_Displacement_from_Acceleration\" rel=\"internal\">3.6 Finding Velocity and Displacement from Acceleration<\/a><\/h2>\r\n<ol class=\"mt-indent-1\" start=\"78\">\r\n \t<li>The acceleration of a particle varies with time according to the equation a(t) = pt<sup>2<\/sup> \u2212 qt<sup>3<\/sup>. Initially, the velocity and position are zero. (a) What is the velocity as a function of time? (b) What is the position as a function of time?<\/li>\r\n \t<li>Between t = 0 and t = t<sub>0<\/sub>, a rocket moves straight upward with an acceleration given by a(t) = A \u2212 Bt<sup>1<\/sup><sup>\/2<\/sup>, where A and B are constants. (a) If x is in meters and t is in seconds, what are the units of A and B? (b) If the rocket starts from rest, how does the velocity vary between t = 0 and t = t<sub>0<\/sub>? (c) If its initial position is zero, what is the rocket\u2019s position as a function of time during this same time interval?<\/li>\r\n \t<li>The velocity of a particle moving along the x-axis varies with time according to v(t) = A + Bt<sup>\u22121<\/sup>, where A = 2 m\/s, B = 0.25 m, and 1.0 s \u2264 t \u2264 8.0 s. Determine the acceleration and position of the particle at t = 2.0 s and t = 5.0 s. Assume that x(t = 1 s) = 0.<\/li>\r\n \t<li>A particle at rest leaves the origin with its velocity increasing with time according to v(t) = 3.2t m\/s. At 5.0 s, the particle\u2019s velocity starts decreasing according to [16.0 \u2013 1.5(t \u2013 5.0)] m\/s. This decrease continues until t = 11.0 s, after which the particle\u2019s velocity remains constant at 7.0 m\/s. (a) What is the acceleration of the particle as a function of time? (b) What is the position of the particle at t = 2.0 s, t = 7.0 s, and t = 12.0 s?<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/section><\/article>&nbsp;","rendered":"<div class=\"textbox textbox--learning-objectives\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Learning Objectives<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<ul>\n<li><span>By the end of this section, you will be able to:<\/span>\n<ul>\n<li>Explain the difference between average velocity and instantaneous velocity.<\/li>\n<li>Describe the difference between velocity and speed.<\/li>\n<li>Calculate the instantaneous velocity given the mathematical equation for the velocity.<\/li>\n<li>Calculate the speed given the instantaneous velocity.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p><span>We have now seen how to calculate the average velocity between two positions. However, since objects in the real world move continuously through space and time, we would like to find the velocity of an object at any single point. We can find the velocity of the object anywhere along its path by using some fundamental principles of calculus. This section gives us better insight into the physics of motion and will be useful in later chapters.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><a class=\"elm-skip-link\" href=\"https:\/\/phys.libretexts.org\/Bookshelves\/University_Physics\/Book%3A_University_Physics_(OpenStax)\/Map%3A_University_Physics_I_-_Mechanics%2C_Sound%2C_Oscillations%2C_and_Waves_(OpenStax)\/3%3A_Motion_Along_a_Straight_Line\/3.2%3A_Instantaneous_Velocity_and_Speed#elm-main-content\" title=\"Press enter to skip to the main content\">Read this online for free &#8211; and legally at Libre Texts . Click here<\/a><\/p>\n<article id=\"elm-main-content\" class=\"elm-content-container\">\n<section class=\"mt-content-container\">\n<div class=\"mt-section\" id=\"section_1\">\n<h1 class=\"editable\">Instantaneous Velocity<\/h1>\n<p>The quantity that tells us how fast an object is moving anywhere along its path is the<strong>instantaneous velocity<\/strong>, usually called simply<em>velocity<\/em>. It is the average velocity between two points on the path in the limit that the time (and therefore the displacement) between the two points approaches zero. To illustrate this idea mathematically, we need to express position x as a continuous function of t denoted by x(t). The expression for the average velocity between two points using this notation is<\/p>\n<p>&nbsp;<\/p>\n<div class=\"MathJax_Display\"><\/div>\n<div class=\"note1\">\n<p class=\"boxtitle\">Instantaneous Velocity<\/p>\n<p>The instantaneous velocity of an object is the limit of the average velocity as the elapsed time approaches zero, or the derivative of x with respect to t:<\/p>\n<div class=\"MathJax_Display\"><\/div>\n<\/div>\n<p>Like average velocity, instantaneous velocity is a vector with dimension of length per time. The instantaneous velocity at a specific time point t<sub>0<\/sub>is the rate of change of the position function, which is the slope of the position function x(t) at t<sub>0<\/sub>. Figure 3.6 shows how the average velocity&#8221;&gt;<span class=\"math\" id=\"MathJax-Span-121\"><span><span class=\"mrow\" id=\"MathJax-Span-122\"><span class=\"texatom\" id=\"MathJax-Span-123\"><span class=\"mrow\" id=\"MathJax-Span-124\"><span class=\"munderover\" id=\"MathJax-Span-125\"><span class=\"mi\" id=\"MathJax-Span-126\">v<\/span><span class=\"mo\" id=\"MathJax-Span-127\">\u00af<\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-128\">=<\/span><span class=\"mfrac\" id=\"MathJax-Span-129\"><span class=\"mrow\" id=\"MathJax-Span-130\"><span class=\"mi\" id=\"MathJax-Span-131\">\u0394<\/span><span class=\"mi\" id=\"MathJax-Span-132\">x<\/span><\/span><span class=\"mrow\" id=\"MathJax-Span-133\"><span class=\"mi\" id=\"MathJax-Span-134\">\u0394<\/span><span class=\"mi\" id=\"MathJax-Span-135\">t<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">v\u00af=\u0394x\u0394t<\/span>between two times approaches the instantaneous velocity at t<sub>0<\/sub>. The instantaneous velocity is shown at time t<sub>0<\/sub>, which happens to be at the maximum of the position function. The slope of the position graph is zero at this point, and thus the instantaneous velocity is zero. At other times, t<sub>1<\/sub>, t<sub>2<\/sub>, and so on, the instantaneous velocity is not zero because the slope of the position graph would be positive or negative. If the position function had a minimum, the slope of the position graph would also be zero, giving an instantaneous velocity of zero there as well. Thus, the zeros of the velocity function give the minimum and maximum of the position function.<\/p>\n<figure><img decoding=\"async\" alt=\"Graph shows position plotted versus time. Position increases from t1 to t2 and reaches maximum at t0. It decreases to at and continues to decrease at t4. The slope of the tangent line at t0 is indicated as the instantaneous velocity.\" class=\"internal default\" width=\"290px\" height=\"233px\" src=\"https:\/\/phys.libretexts.org\/@api\/deki\/files\/3708\/clipboard_efc61ece1c38ff1f7239a6cabb1619e94?revision=1&amp;size=bestfit&amp;width=290&amp;height=233\" \/><figcaption>Figure<em>In a graph of position versus time, the instantaneous velocity is the slope of the tangent line at a given point. The average velocities.\u00a0<span class=\"MJX_Assistive_MathML\" role=\"presentation\">\u0394<\/span>t \u2192 0, the average velocity approaches the instantaneous velocity at t = t<sub>0<\/sub>.<\/em><\/figcaption><\/figure>\n<div class=\"example\">\n<p class=\"boxtitle\">Example 3.2: Finding Velocity from a Position-Versus-Time Graph<\/p>\n<p>Given the position-versus-time graph of Figure 3.7, find the velocity-versus-time graph.<\/p>\n<figure><img decoding=\"async\" alt=\"Graph shows position in kilometers plotted as a function of time at minutes. It starts at the origin, reaches 0.5 kilometers at 0.5 minutes, remains constant between 0.5 and 0.9 minutes, and decreases to 0 at 2.0 minutes.\" class=\"internal default\" width=\"307px\" height=\"274px\" src=\"https:\/\/phys.libretexts.org\/@api\/deki\/files\/3709\/clipboard_efaca94677e552a40cfa727e5283ce0bb?revision=1&amp;size=bestfit&amp;width=307&amp;height=274\" \/><figcaption><em>The object starts out in the positive direction, stops for a short time, and then reverses direction, heading back toward the origin. Notice that the object comes to rest instantaneously, which would require an infinite force. Thus, the graph is an approximation of motion in the real world. (The concept of force is discussed in Newton\u2019s Laws of Motion.)<\/em><\/figcaption><\/figure>\n<div class=\"mt-section\" id=\"section_2\">\n<p>&nbsp;<\/p>\n<h3 class=\"editable\">Strategy<\/h3>\n<p>The graph contains three straight lines during three time intervals. We find the velocity during each time interval by taking the slope of the line using the grid.<\/p>\n<\/div>\n<div class=\"mt-section\" id=\"section_3\">\n<p>&nbsp;<\/p>\n<h3 class=\"editable\">Solution<\/h3>\n<p>The graph of these values of velocity versus time is shown in Figure 3.8.<\/p>\n<figure><img decoding=\"async\" alt=\"Graph shows velocity in meters per second plotted as a function of time at seconds. The velocity is 1 meter per second between 0 and 0.5 seconds, zero between 0.5 and 1.0 seconds, and -0.5 between 1.0 and 2.0 seconds.\" class=\"internal default\" width=\"298px\" height=\"212px\" src=\"https:\/\/phys.libretexts.org\/@api\/deki\/files\/3710\/clipboard_e7ce331edf68d0702a995c9fe676d5969?revision=1&amp;size=bestfit&amp;width=298&amp;height=212\" \/><figcaption>Figure<em>The velocity is positive for the first part of the trip, zero when the object is stopped, and negative when the object reverses direction.<\/em><\/figcaption><\/figure>\n<\/div>\n<div class=\"mt-section\" id=\"section_4\">\n<p>&nbsp;<\/p>\n<h3 class=\"editable\">Significance<\/h3>\n<p>During the time interval between 0 s and 0.5 s, the object\u2019s position is moving away from the origin and the position-versus-time curve has a positive slope. At any point along the curve during this time interval, we can find the instantaneous velocity by taking its slope, which is +1 m\/s, as shown in Figure 3.8. In the subsequent time interval, between 0.5 s and 1.0 s, the position doesn\u2019t change and we see the slope is zero. From 1.0 s to 2.0 s, the object is moving back toward the origin and the slope is \u22120.5 m\/s. The object has reversed direction and has a negative velocity.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"mt-section\" id=\"section_5\">\n<p>&nbsp;<\/p>\n<h1 class=\"editable\">Speed<\/h1>\n<p>In everyday language, most people use the terms speed and velocity interchangeably. In physics, however, they do not have the same meaning and are distinct concepts. One major difference is that speed has no direction; that is, speed is a scalar.<\/p>\n<p>We can calculate the\u00a0<strong>average speed\u00a0<\/strong>by finding the total distance traveled divided by the elapsed time:<\/p>\n<\/div>\n<\/section>\n<p><strong>Table 3.1 Speeds of various objects in metres per second<\/strong><\/p>\n<\/article>\n<p>\u00a0 \u00a0 \u00a0 \u00a0 Continental drift = 10 <sup>-7<\/sup> m\/s<\/p>\n<p>Brisk walk = 1.7 m\/s<\/p>\n<p>Cyclist = 4. 4 m\/s<\/p>\n<p>Sprint runner = 12.2 m\/s<\/p>\n<p>Official land speed record = 341.4 m\/s<\/p>\n<p>Speed of sound = 343 m\/s<\/p>\n<p>Escape velocity =\u00a0Escape velocity is the velocity at which an object must be launched so that it overcomes Earth\u2019s gravity and is not pulled back toward Earth = 11,200 m\/s<\/p>\n<p>ISS = International Space Station =. 7,600 \u00a0m\/s<\/p>\n<p>Orbital speed of the Earth =.\u00a029,700 m\/s<\/p>\n<p>Speed of light in.a vacuum = 3.00 x 10<sup>8<\/sup> m\/s<\/p>\n<p>&nbsp;<\/p>\n<article id=\"elm-main-content\" class=\"elm-content-container\">\n<section class=\"mt-content-container\">\n<div class=\"mt-section\" id=\"section_7\">\n<h1 class=\"editable\">Calculating Instantaneous Velocity<\/h1>\n<p>When calculating instantaneous velocity, we need to specify the explicit form of the position function x(t). For the moment, let\u2019s use polynomials x(t) = At<sup>n<\/sup>, because they are easily differentiated using the power rule of calculus:<\/p>\n<div class=\"MathJax_Display\"><\/div>\n<p>The following example illustrates the use of Equation 3.7.<\/p>\n<div class=\"example\">\n<p class=\"boxtitle\">Example 3.3: Instantaneous Velocity Versus Average Velocity<\/p>\n<p>The position of a particle is given by x(t) = 3.0t + 0.5t<sup>3<\/sup>m.<\/p>\n<ol start=\"1\">\n<li>Using Equation 3.4 and Equation 3.7, find the instantaneous velocity at t = 2.0 s.<\/li>\n<li>Calculate the average velocity between 1.0 s and 3.0 s.<\/li>\n<\/ol>\n<div class=\"mt-section\" id=\"section_8\">\n<p>&nbsp;<\/p>\n<h3 class=\"editable\">Strategy<\/h3>\n<p>Equation 3.4 give the instantaneous velocity of the particle as the derivative of the position function. Looking at the form of the position function given, we see that it is a polynomial in t. Therefore, we can use Equation 3.7, the power rule from calculus, to find the solution. We use Equation 3.6 to calculate the average velocity of the particle.<\/p>\n<\/div>\n<div class=\"mt-section\" id=\"section_9\">\n<h3 class=\"editable\">Solution<\/h3>\n<ol start=\"1\">\n<li><span class=\"math\" id=\"MathJax-Span-457\"><span><span class=\"mrow\" id=\"MathJax-Span-458\"><span class=\"mfrac\" id=\"MathJax-Span-459\"><span class=\"mrow\" id=\"MathJax-Span-460\"><span class=\"mi\" id=\"MathJax-Span-461\">\u00a0<span class=\"ql-right-eqno\"> &nbsp; <\/span><span class=\"ql-left-eqno\"> &nbsp; <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/physicsforlifesciences1phys1108\/wp-content\/ql-cache\/quicklatex.com-e7c321c421f31d9623bda105252fd702_l3.png\" height=\"38\" width=\"631\" class=\"ql-img-displayed-equation quicklatex-auto-format\" alt=\"&#92;&#91;&#92;&#102;&#114;&#97;&#99;&#32;&#123;&#100;&#120;&#125;&#123;&#100;&#116;&#125;&#61;&#118;&#40;&#116;&#41;&#60;&#47;&#115;&#112;&#97;&#110;&#62;&#60;&#47;&#115;&#112;&#97;&#110;&#62;&#60;&#47;&#115;&#112;&#97;&#110;&#62;&#60;&#47;&#115;&#112;&#97;&#110;&#62;&#60;&#47;&#115;&#112;&#97;&#110;&#62;&#60;&#47;&#115;&#112;&#97;&#110;&#62;&#32;&#32;&#47;&#102;&#114;&#97;&#99;&#92;&#93;\" title=\"Rendered by QuickLaTeX.com\" \/><\/span><\/span><\/span><\/span><\/span><\/span><\/li>\n<li>\u00a0dx\/dt = v(t) = 3.0 + 1.5t<sup>2<\/sup>\u00a0<sup>\u00a0<\/sup>m\/s<\/li>\n<li>Substituting t = 2.0 s into this equation gives v<sub>(2.0 s)<\/sub> = [3.0 + 1.5(2.0)<sup>2<\/sup>] m\/s = 9.0 m\/s.<\/li>\n<li>To determine the average velocity of the particle between 1.0 s and 3.0 s, we calculate the values of x(1.0 s) and x(3.0 s):<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"example\">\n<p>&nbsp;<\/p>\n<p class=\"boxtitle\">Example 3.4: Instantaneous Velocity Versus Speed<\/p>\n<p>Consider the motion of a particle in which the position is x(t) = 3.0t \u2212 3t<sup>2<\/sup>m.<\/p>\n<ol start=\"1\">\n<li>What is the instantaneous velocity at t = 0.25 s, t = 0.50 s, and t = 1.0 s?<\/li>\n<li>What is the speed of the particle at these times?<\/li>\n<\/ol>\n<div class=\"mt-section\" id=\"section_11\">\n<p>&nbsp;<\/p>\n<h3 class=\"editable\">Strategy<\/h3>\n<p>The instantaneous velocity is the derivative of the position function and the speed is the magnitude of the instantaneous velocity. We use Equation 3.4 and Equation 3.7 to solve for instantaneous velocity.<\/p>\n<\/div>\n<div class=\"mt-section\" id=\"section_12\">\n<p>&nbsp;<\/p>\n<h3 class=\"editable\">Solution<\/h3>\n<ol start=\"1\">\n<li><\/li>\n<li>v(0.25 s) = 1.50 m\/s, v(0.5 s) = 0 m\/s, v(1.0 s) = \u22123.0 m\/s<\/li>\n<li>Speed = |v(t)| = 1.50 m\/s, 0.0 m\/s, and 3.0 m\/s<\/li>\n<\/ol>\n<\/div>\n<div class=\"mt-section\" id=\"section_13\">\n<p>&nbsp;<\/p>\n<h3 class=\"editable\">Significance<\/h3>\n<p>The velocity of the particle gives us direction information, indicating the particle is moving to the left (west) or right (east). The speed gives the magnitude of the velocity. By graphing the position, velocity, and speed as functions of time, we can understand these concepts visually Figure 3.9. In (a), the graph shows the particle moving in the positive direction until t = 0.5 s, when it reverses direction. The reversal of direction can also be seen in (b) at 0.5 s where the velocity is zero and then turns negative. At 1.0 s it is back at the origin where it started. The particle\u2019s velocity at 1.0 s in (b) is negative, because it is traveling in the negative direction. But in (c), however, its speed is positive and remains positive throughout the travel time. We can also interpret velocity as the slope of the position-versus-time graph. The slope of x(t) is decreasing toward zero, becoming zero at 0.5 s and increasingly negative thereafter. This analysis of comparing the graphs of position, velocity, and speed helps catch errors in calculations. The graphs must be consistent with each other and help interpret the calculations.<\/p>\n<figure><img decoding=\"async\" alt=\"Graph A shows position in meters plotted versus time in seconds. It starts at the origin, reaches maximum at 0.5 seconds, and then start to decrease crossing x axis at 1 second. Graph B shows velocity in meters per second plotted as a function of time at seconds. Velocity linearly decreases from the left to the right. Graph C shows absolute velocity in meters per second plotted as a function of time at seconds. Graph has a V-leeter shape. Velocity decreases till 0.5 seconds; then it starts to increase.\" class=\"internal default\" width=\"596px\" height=\"182px\" src=\"https:\/\/phys.libretexts.org\/@api\/deki\/files\/3732\/clipboard_eba5e0c299a72d21ee1326256584794aa?revision=1&amp;size=bestfit&amp;width=596&amp;height=182\" \/><figcaption>Figure<em>(a) Position: x(t) versus time. (b) Velocity: v(t) versus time. The slope of the position graph is the velocity. A rough comparison of the slopes of the tangent lines in (a) at 0.25 s, 0.5 s, and 1.0 s with the values for velocity at the corresponding times indicates they are the same values. (c) Speed: |v(t)| versus time. Speed is always a positive number.<\/em><\/figcaption><\/figure>\n<\/div>\n<\/div>\n<div class=\"exercise\">\n<p class=\"boxtitle\">Check Your Understanding 3.2<\/p>\n<p>The position of an object as a function of time is x(t) = \u22123t<sup>2<\/sup>m. (a) What is the velocity of the object as a function of time? (b) Is the velocity ever positive? (c) What are the velocity and speed at t = 1.0 s?<\/p>\n<\/div>\n<div>\n<div class=\"mt-section\" id=\"section_14\">\n<p>&nbsp;<\/p>\n<h1 class=\"editable\">Contributors<\/h1>\n<\/div>\n<\/div>\n<ul>\n<li>\n<section id=\"main-content\" data-is-baked=\"true\">\n<p id=\"eip-677\">Samuel J. Ling (Truman State University),\u00a0Jeff Sanny (Loyola Marymount University), and Bill Moebs\u00a0with many contributing authors. This work is licensed by OpenStax University Physics under a\u00a0<a href=\"http:\/\/creativecommons.org\/licenses\/by\/4.0\/\" rel=\"external nofollow noopener\" target=\"_blank\" class=\"external\">Creative Commons Attribution License (by 4.0)<\/a>.<\/p>\n<\/section>\n<h1>Exercises<\/h1>\n<h2 class=\"editable\"><a title=\"3.2: Instantaneous Velocity and Speed\" href=\"https:\/\/phys.libretexts.org\/Bookshelves\/University_Physics\/Book%3A_University_Physics_(OpenStax)\/Map%3A_University_Physics_I_-_Mechanics%2C_Sound%2C_Oscillations%2C_and_Waves_(OpenStax)\/3%3A_Motion_Along_a_Straight_Line\/3.2%3A_Instantaneous_Velocity_and_Speed\" rel=\"internal\">Instantaneous Velocity and Speed<\/a><\/h2>\n<ol class=\"mt-indent-1\" start=\"30\">\n<li>A woodchuck runs 20 m to the right in 5 s, then turns and runs 10 m to the left in 3 s. (a) What is the average velocity of the woodchuck? (b) What is its average speed?<\/li>\n<li>Sketch the velocity-versus-time graph from the following position-versus-time graph.<\/li>\n<\/ol>\n<p><img decoding=\"async\" alt=\"Graph shows position in meters plotted versus time in seconds. It starts at the origin, reaches 4 meters at 0.4 seconds; decreases to -2 meters at 0.6 seconds, reaches minimum of -6 meters at 1 second, increases to -4 meters at 1.6 seconds, and reaches 2 meters at 2 seconds.\" class=\"internal default\" width=\"299px\" height=\"276px\" src=\"https:\/\/phys.libretexts.org\/@api\/deki\/files\/3764\/clipboard_e9067e449d507cb185bcd2db18f0f8378?revision=1&amp;size=bestfit&amp;width=299&amp;height=276\" \/><\/p>\n<ol class=\"mt-indent-1\" start=\"32\">\n<li>Sketch the velocity-versus-time graph from the following position-versus-time graph.<\/li>\n<\/ol>\n<p><img decoding=\"async\" alt=\"Graph shows position plotted versus time in seconds. Graph has a sinusoidal shape. It starts with the positive value at zero time, changes to negative, and then starts to increase.\" class=\"internal default\" width=\"287px\" height=\"204px\" src=\"https:\/\/phys.libretexts.org\/@api\/deki\/files\/3765\/clipboard_e8f2d6f0c158a7dbb0a6473461e708e4a?revision=1&amp;size=bestfit&amp;width=287&amp;height=204\" \/><\/p>\n<ol class=\"mt-indent-1\" start=\"33\">\n<li>Given the following velocity-versus-time graph, sketch the position-versus-time graph.<\/li>\n<\/ol>\n<p><img decoding=\"async\" alt=\"Graph shows velocity plotted versus time. It starts with the positive value at zero time, decreases to the negative value and remains constant.\" class=\"internal default\" width=\"291px\" height=\"215px\" src=\"https:\/\/phys.libretexts.org\/@api\/deki\/files\/3766\/clipboard_eff678ded18eed80b84f692f0c227a6cc?revision=1&amp;size=bestfit&amp;width=291&amp;height=215\" \/><\/p>\n<ol class=\"mt-indent-1\" start=\"34\">\n<li>An object has a position function x(t) = 5t m. (a) What is the velocity as a function of time? (b) Graph the position function and the velocity function. 35. A particle moves along the x-axis according to x(t) = 10t \u2212 2t<sup>2<\/sup> m. (a) What is the instantaneous velocity at t = 2 s and t = 3 s? (b) What is the instantaneous speed at these times? (c) What is the average velocity between t = 2 s and t = 3 s?<\/li>\n<li><strong>Unreasonable results<\/strong>. A particle moves along the x-axis according to x(t) = 3t<sup>3<\/sup> + 5t . At what time is the velocity of the particle equal to zero? Is this reasonable?<\/li>\n<\/ol>\n<\/li>\n<li>\n<h2 class=\"editable\"><a title=\"3.6: Finding Velocity and Displacement from Acceleration\" href=\"https:\/\/phys.libretexts.org\/Bookshelves\/University_Physics\/Book%3A_University_Physics_(OpenStax)\/Map%3A_University_Physics_I_-_Mechanics%2C_Sound%2C_Oscillations%2C_and_Waves_(OpenStax)\/3%3A_Motion_Along_a_Straight_Line\/3.6%3A_Finding_Velocity_and_Displacement_from_Acceleration\" rel=\"internal\">3.6 Finding Velocity and Displacement from Acceleration<\/a><\/h2>\n<ol class=\"mt-indent-1\" start=\"78\">\n<li>The acceleration of a particle varies with time according to the equation a(t) = pt<sup>2<\/sup> \u2212 qt<sup>3<\/sup>. Initially, the velocity and position are zero. (a) What is the velocity as a function of time? (b) What is the position as a function of time?<\/li>\n<li>Between t = 0 and t = t<sub>0<\/sub>, a rocket moves straight upward with an acceleration given by a(t) = A \u2212 Bt<sup>1<\/sup><sup>\/2<\/sup>, where A and B are constants. (a) If x is in meters and t is in seconds, what are the units of A and B? (b) If the rocket starts from rest, how does the velocity vary between t = 0 and t = t<sub>0<\/sub>? (c) If its initial position is zero, what is the rocket\u2019s position as a function of time during this same time interval?<\/li>\n<li>The velocity of a particle moving along the x-axis varies with time according to v(t) = A + Bt<sup>\u22121<\/sup>, where A = 2 m\/s, B = 0.25 m, and 1.0 s \u2264 t \u2264 8.0 s. Determine the acceleration and position of the particle at t = 2.0 s and t = 5.0 s. Assume that x(t = 1 s) = 0.<\/li>\n<li>A particle at rest leaves the origin with its velocity increasing with time according to v(t) = 3.2t m\/s. At 5.0 s, the particle\u2019s velocity starts decreasing according to [16.0 \u2013 1.5(t \u2013 5.0)] m\/s. This decrease continues until t = 11.0 s, after which the particle\u2019s velocity remains constant at 7.0 m\/s. (a) What is the acceleration of the particle as a function of time? (b) What is the position of the particle at t = 2.0 s, t = 7.0 s, and t = 12.0 s?<\/li>\n<\/ol>\n<\/li>\n<\/ul>\n<\/div>\n<\/section>\n<\/article>\n<p>&nbsp;<\/p>\n","protected":false},"author":9,"menu_order":10,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-807","chapter","type-chapter","status-publish","hentry"],"part":113,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/physicsforlifesciences1phys1108\/wp-json\/pressbooks\/v2\/chapters\/807","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/physicsforlifesciences1phys1108\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/physicsforlifesciences1phys1108\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/physicsforlifesciences1phys1108\/wp-json\/wp\/v2\/users\/9"}],"version-history":[{"count":25,"href":"https:\/\/pressbooks.bccampus.ca\/physicsforlifesciences1phys1108\/wp-json\/pressbooks\/v2\/chapters\/807\/revisions"}],"predecessor-version":[{"id":921,"href":"https:\/\/pressbooks.bccampus.ca\/physicsforlifesciences1phys1108\/wp-json\/pressbooks\/v2\/chapters\/807\/revisions\/921"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/physicsforlifesciences1phys1108\/wp-json\/pressbooks\/v2\/parts\/113"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/physicsforlifesciences1phys1108\/wp-json\/pressbooks\/v2\/chapters\/807\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/physicsforlifesciences1phys1108\/wp-json\/wp\/v2\/media?parent=807"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/physicsforlifesciences1phys1108\/wp-json\/pressbooks\/v2\/chapter-type?post=807"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/physicsforlifesciences1phys1108\/wp-json\/wp\/v2\/contributor?post=807"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/physicsforlifesciences1phys1108\/wp-json\/wp\/v2\/license?post=807"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}