A particle moves along a straight line with velocity on the interval
MATHEMATICS · Single correct · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
A particle moves along a straight line with velocity v(t) on the interval 0≤t≤10. Its net displacement on this interval is −6 meters, while the total distance it travels is 14 meters. Which of the following statements best explains how these two quantities can have different magnitudes?
(a) Distance is found by integrating the speed and then dividing by the length of the interval, whereas displacement is the integral of velocity without that division, so they need not be equal.
(b) On part of the interval the particle’s speed decreases to zero; these rest periods reduce displacement but leave distance unchanged, so the two magnitudes differ.
(c) The velocity must change sign on [0,10], so motion in opposite directions partially cancels when computing displacement.
(d) Because the acceleration is negative for part of the interval, both distance and displacement drop, but displacement drops more, making its magnitude smaller than the distance traveled.