Chapter_3 - Chapter 3 Motion in Two and Three Dimensions...

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171 Chapter 3 Motion in Two and Three Dimensions Conceptual Problems 1 [SSM] Can the magnitude of the displacement of a particle be less than the distance traveled by the particle along its path? Can its magnitude be more than the distance traveled? Explain. Determine the Concept The distance traveled along a path can be represented as a sequence of displacements. Suppose we take a trip along some path and consider the trip as a sequence of many very small displacements. The net displacement is the vector sum of the very small displacements, and the total distance traveled is the sum of the magnitudes of the very small displacements. That is, total distance = N N 1, 3 , 2 2 , 1 1 , 0 ... Δ + + Δ + Δ + Δ r r r r r r r r where N is the number of very small displacements. (For this to be exactly true we have to take the limit as N goes to infinity and each displacement magnitude goes to zero.) Now, using the shortest distance between two points is a straight line, we have N N N 1, 3 , 2 2 , 1 1 , 0 , 0 ... Δ + + Δ + Δ + Δ Δ r r r r r r r r r r , where N , 0 r r Δ is the magnitude of the net displacement. Hence, we have shown that the magnitude of the displacement of a particle is less than or equal to the distance it travels along its path. 2 Give an example in which the distance traveled is a significant amount, yet the corresponding displacement is zero. Can the reverse be true? If so, give an example.
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Chapter 3 172 Determine the Concept The displacement of an object is its final position vector minus its initial position vector ( i f r r r r r r = Δ ). The displacement can be less but never more than the distance traveled. Suppose the path is one complete trip around Earth at the equator. Then, the displacement is 0 but the distance traveled is 2 π R e . No, the reverse cannot be true. 3 What is the average velocity of a batter who hits a home run (from when he hits the ball to when he touches home plate after rounding the bases)? Determine the Concept The important distinction here is that average velocity is being requested, as opposed to average speed . The average velocity is defined as the displacement divided by the elapsed time. 0 0 av = Δ = Δ Δ = t t r v r r The displacement for any trip around the bases is zero. Thus we see that no matter how fast the runner travels, the average velocity is always zero at the end of each complete circuit of the bases. What is the correct answer if we were asked for average speed ? The average speed is defined as the distance traveled divided by the elapsed time. t v Δ distance total av = For one complete circuit of any track, the total distance traveled will be greater than zero and so the average speed is not zero. 4 A baseball is hit so its initial velocity upon leaving the bat makes an angle of 30 ° above the horizontal. It leaves that bat at a height of 1.0 m above the ground and lands untouched for a single. During its flight, from just after it leaves the bat to just before it hits the ground, describe how the angle between its velocity and acceleration vectors changes.
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Chapter_3 - Chapter 3 Motion in Two and Three Dimensions...

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