An object moves along a straight line so that its velocity (in meters per second)…

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

Problem

An object moves along a straight line so that its velocity

v(t)

(in meters per second) satisfies the differential equation

\dfrac{dv}{dt}=t^{2}-4

for

t\ge 0

. At time

t=0

, the object’s velocity is

6\text{ m/s}

, and its position is

10\text{ m}

. Complete all parts. Include units in your answers and justify each step. (a) Determine

v(t)

for

t\ge 0

that satisfies the given condition. (b) Using the result from part (a), find the position function

s(t)

that satisfies the initial position condition. (c) Find all values of

t

in the interval

0< t\le 3

at which the object is momentarily at rest, and classify each as a local maximum or minimum of

s(t)

with justification. (d) Discuss any restrictions on the domain of

v(t)

and

s(t)

that arise from your integrations, and interpret these restrictions in the physical context of the problem.