A particle moves along a line

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

Problem

A particle moves along a line. Its position, in meters, is modeled by

s(u)=2u^{3}-3u

where

u

is a parameter related to time by

t=u^{2}

seconds. The interval

1\le t\le 9

corresponds to

1\le u\le 3

. (a) Express the particle’s position as an explicit function of time

t

for

1\le t\le 9

. (b) Derive an expression for the velocity

v(t)

in terms of

t

on this interval. (c) Compute the particle’s average velocity for

1\le t\le 9

seconds and state the units. (d) Determine whether there exists a time

c

(1,9)

at which the instantaneous velocity equals the average velocity found in part (c). Provide justification for your conclusion.