A particle moves along a straight line with velocity (in meters per second) given by…
MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation
Problem
A particle moves along a straight line with velocity (in meters per second) given by the piecewise-defined function
For 0≤t≤4: v(t)=3−t
For 4<t≤7: v(t)=t−5
Time t is measured in seconds, and t=0 corresponds to the instant the particle starts moving.
Throughout this problem, the term "displacement" refers to the net change in position, while "distance traveled" refers to the total accumulated magnitude of movement.
(a) Identify the subintervals of [0,7] on which the velocity is positive and those on which it is negative. Explain your reasoning.
(b) Describe, in words, the difference between the particle’s displacement and the distance it travels on the interval [0,7]. Include units in your description.
(c) Without performing any explicit integration, determine whether the magnitude of the displacement over [0,7] is less than, equal to, or greater than the total distance traveled over the same interval. Justify your conclusion by referencing geometric features of the velocity graph.