Distance traveled physics calculator7/3/2023 For what values of t is the position function s increasing? Explain why this is the case using relevant information about the velocity function v.Īrea under the graph of the velocity function In Preview Activity 4.1, we encountered a fundamental fact: when a moving object’s velocity is constant (and positive), the area under the velocity curve over a given interval tells us the distance the object traveled.On the right-hand axes provided in Figure 4.1, sketch a labeled graph of the position function y = s(t).Find an algebraic formula, s(t), for the position of the person at time t, assuming that s(0) = 0.How far did the person travel during the two hours? How is this distance related to the area of a certain region under the graph of y = v(t)?.The right-hand axes 209 will be used in question (d). Note that while the scale on the two sets of axes is the same, the units on the right-hand axes differ from those on the left. On the left-hand axes provided in Figure 4.1, sketch a labeled graph of the velocity function v(t) = 3.Suppose that a person is taking a walk along a long straight path and walks at a constant rate of 3 miles per hour.įigure 4.1: At left, axes for plotting y = v(t) at right, for plotting y = s(t). It is this particular problem that will be the focus of our attention in most of Chapter 4: if we know the instantaneous rate of change of a function, are we able to determine the function itself? To begin, we start with a more focused question: if we know the instantaneous velocity of an object moving along a straight line path, can we determine its corresponding position function? In a much smaller number of situations so far, we have encountered the reverse situation where we instead know the derivative, f 0, and have tried to deduce information about f. That is, we have typically proceeded from a function f to its derivative, f 0, and then used the meaning of the derivative to help us answer important questions. The vast majority of the problems and applications we have considered have involved the situation where a particular function is known and we seek information that relies on knowing the function’s instantaneous rate of change. Given a differentiable function f, we are now able to find its derivative and use this derivative to determine the function’s instantaneous rate of change at any point in the domain, as well as to find where the function is increasing or decreasing, is concave up or concave down, and has relative extremes. Moreover, from this foundational problem involving position and velocity we have learned a great deal. Thus, given a differentiable position function, we are able to know the exact velocity of the moving object at any point in time. From this starting point, we investigated the average velocity of the ball on a given interval, computed by the difference quotient s(b)−s(a) b−a, and eventually found that we could determine the exact instantaneous velocity of the ball at time t by taking the derivative of 207 208 the position function, s 0 (t) = lim h→0 s(t h) − s(t) h. In particular, we stipulated that a tennis ball tossed into the air had its height s (in feet) at time t (in seconds) given by s(t) = 64 − 16(t − 1) 2. In the very first section of the text, we considered a situation where a moving object had a known position at time t. If velocity is negative, how does this impact the problem of finding distance traveled?. ![]()
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