A particle moves in the plane so that its position is given by for

MATHEMATICS · Multi-part · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions

Problem

A particle moves in the plane so that its position is given by

\mathbf r(t)=\langle t^{3}-6t^{2}+9t,\,\tfrac{1}{2}t^{2}\rangle

for

0\le t\le 4

, where

t

is in seconds and the components are in meters. (a) Determine the velocity vector

\mathbf v(t)

and the speed

s(t)

of the particle. (b) Set up—but do not evaluate—an expression involving one or more definite integrals that gives the total distance the particle travels on

0\le t\le 4

. Be sure to account for any changes in direction. (c) Without computing the integral(s), explain how you would identify each sub-interval on which the integrand must be adjusted to ensure the integrand represents speed rather than signed velocity. (d) Suppose numerical evaluation of the integral(s) in part (b) yields

12.7

meters, while direct evaluation of

\mathbf r(4)-\mathbf r(0)

gives a displacement of

4

meters. Interpret these two results in the context of the particle’s motion.

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