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Velocity Progear ROGUE PB 9.0 SERVICE BAG, Black

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Suppose that a person is walking in such a way that her velocity varies slightly according to the information given in the table below and graph given in Figure 4.4. Answer the same questions as in (c) and (d) but instead using the interval [0, 1]. (f) What is the value of s(2) − s(0)? What does this result tell you about the flight of the ball? How is this value connected to the provided graph of y = v(t)? Explain. C

BL] [OL] Before students read the section, ask them to give examples of ways they have heard the word speed used. Then ask them if they have heard the word velocity used. Explain that these words are often used interchangeably in everyday life, but their scientific definitions are different. Tell students that they will learn about these differences as they read the section. In the previous chapter we found the instantaneous velocity by calculating the derivative of the position function with respect to time. We can do the same operation in two and three dimensions, but we use vectors. The instantaneous velocity vector is now Suppose that an object moving along a straight line path has its velocity v (in meters per second) at time t (in seconds) given by the piecewise linear function whose graph is pictured in Figure 4.8. We view movement to the right as being in the positive direction (with positive velocity), while movement to the left is in the negative direction. Suppose We find that the travel time before A meets B is 2329.5 seconds (seems like a massive number, but it is, after all, equal to ~39 minutes). Using the graph of y = v(t) provided in Figure 4.6, find the exact area of the region under the velocity curve between t = 1 2 and t = 1. What is the meaning of the value you find?For what values of t is the velocity of the ball positive? What does this tell you about the motion of the ball on this interval of time values? It is with a heavy heart and much consideration we have decided that loadout, as we all know it, will cease trading with immediate effect. How could you get a better approximation of the distance traveled on [0, 2]? Explain, and then find this new estimate.

There is more to motion than distance and displacement. Questions such as, “How long does a foot race take?” and “What was the runner’s speed?” cannot be answered without an understanding of other concepts. In this section we will look at time, speed, and velocity to expand our understanding of motion. When the velocity of a moving object is positive, the object’s position is always increasing. While we will soon consider situations where velocity is negative and think about the ramifications of this condition on distance traveled, for now we continue to assume that we are working with a positive velocity function. In that setting, we have established that whenever v is actually constant, the exact distance traveled on an interval is the area under the velocity curve; furthermore, we have observed that when v is not constant, we can estimate the total distance traveled by finding the areas of rectangles that help to approximate the area under the velocity curve on the given interval. Hence, we see the importance of the problem of finding the area between a curve and the horizontal axis: besides being an interesting geometric question, in the setting of the curve being the (positive) velocity of a moving object, the area under the curve over an interval tells us the exact distance traveled on the interval. We can estimate this area any time we have a graph of the velocity function or a table of data that tells us some relevant values of the function. In Activity 4.1, we also encountered an alternate approach to finding the distance traveled. In particular, if we know a formula for the instantaneous velocity, y = v(t), of the moving body at time t, then we realize that v must be the derivative of some corresponding position function s. If we can find a formula for s from the formula for v, it follows that we know the position of the object at time t. In addition, under the assumption that velocity is positive, the change in position over a given interval then tells us the distance traveled on that interval. For a simple example, consider the situation from Preview Activity 4.1, where a person is walking along a straight line and has velocity function v(t) = 3 mph. As pictured in Use standard gravity, a = 9.80665 m/s 2, for equations involving the Earth's gravitational force as the acceleration rate of an object. Solve more complex problems. If an object turns or changes speed, don't get confused. Average velocity is still calculated only from the total displacement, and the total time. It doesn't matter what happens in between the start point. Here are a few examples of journeys with the exact same displacement and time, and therefore the same average velocity: Determine the total distance traveled and the total change in position on the time interval 0 ≤ t ≤ 2. What is the object’s position at t = 2?Bart walks west at 5 m/s for 3 seconds, then turns around and walks east at 7 m/s for 1 second. We can treat the eastward movement as "negative movement west," so total displacement = (5 m/s west)(3 s) + (-7 m/s west)(1 s) = 8 meters. Total time = 4s. Average velocity = 8 m west / 4s = 2 m/s west. 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.

Now suppose that you know that v is given by v(t) = 0.5t 3 − 1.5t 2 + 1.5t + 1.5. Remember that v is the derivative of the walker’s position function, s. Find a formula for s so that s 0 = v. Ok, the best way to have a swing at this, is to first of all, ignore the flagpole, and deal with that part of the problem afterwards, let's first find out when they cross paths, and from there we can figure out how far each has traveled and therefore find their relative distances from the flagpole.Area 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. As seen at left in Figure 4.2, if we consider an object

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