A particle moves along a straight line so that its velocity (in meters per second) at…

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

Problem

A particle moves along a straight line so that its velocity

v

(in meters per second) at time

t

seconds is given by

v(t)=\begin{cases} 4-2t, & 0\le t<2,\\ -t+1, & 2\le t\le 4. \end{cases}

Assume

t=0

corresponds to the instant the particle passes the origin. (a) Identify all time intervals on

0\le t\le 4

during which the particle is moving in the positive direction and all intervals during which it is moving in the negative direction. Explain how you determined these intervals from the definition of

v(t)

. (b) Let

s(t)

denote the position of the particle measured from the origin. Explain why the definite integral

\int_a^b v(t)\,dt

gives the net change

s(b)-s(a)

, and interpret the sign of this quantity in the context of the motion. (c) Without evaluating any integrals, determine whether the net displacement of the particle over

0\le t\le 4

is positive, negative, or zero. Justify your answer verbally by comparing areas under and above the

t

–axis. (d) Suppose that

A_+

denotes the total area of the regions where the graph of

v

is above the

t

–axis on

0\le t\le 4

, and

A_-

denotes the total area of the regions where the graph of

v

is below the

t

–axis on the same interval. Explain why the total distance traveled is

A_+ + A_-

. Then state, without calculation, which of

A_+

A_-

is larger and what that implies about the total distance compared with the magnitude of the net displacement.

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