A car travels along a straight road with velocity meters per second for

MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration

Problem

A car travels along a straight road with velocity

v(t)=t^{3}-6t^{2}+9t

meters per second for

0\le t\le5

, where

t

is measured in seconds. The car’s position at time

t=0

s(0)=0

meters. (a) Determine the car’s displacement on the interval

0\le t\le5

. (b) Compute the total distance the car travels on the interval

0\le t\le5

. Show how you arrive at your answer. (c) At time

t=2

seconds the car is 25 meters east of the origin. Define a position function

p(t)=\displaystyle\int_{2}^{t}v(u)\,du+25

. Find

p(5)

and interpret the result. (d) Identify the time intervals on

0\le t\le5

during which the car is moving to the east and those during which it is moving to the west. Justify your answer thoroughly.