3.4 EXERCISES
For the following exercises, the given functions represent
the position of a particle traveling along a horizontal line.
a. Find the velocity and acceleration functions.
b. Determine the time intervals when the object is
slowing down or speeding up.
150. $s(t) = 2t^3 – 3t^2 – 12t + 8$
151. $s(t) = 2t^3 – 15t^2 + 36t – 10$
152. $s(t) = \frac{t}{1+t^2}$
153. A rocket is fired vertically upward from the ground.
The distance $s$ in feet that the rocket travels from the
ground after $t$ seconds is given by $s(t) = -16t^2 + 560t$.
a. Find the velocity of the rocket 3 seconds after being
fired.
b. Find the acceleration of the rocket 3 seconds after
being fired.
154. A ball is thrown downward with a speed of 8 ft/
s from the top of a 64-foot-tall building. After $t$ seconds,
its height above the ground is given by
$s(t) = -16t^2 - 8t + 64$.
a. Determine how long it takes for the ball to hit the
ground.
b. Determine the velocity of the ball when it hits the
ground.
155. The position function $s(t) = t^2 – 3t – 4$ represents
the position of the back of a car backing out of a driveway
and then driving in a straight line, where $s$ is in feet and
$t$ is in seconds. In this case, $s(t) = 0$ represents the time
at which the back of the car is at the garage door, so
$s(0) = -4$ is the starting position of the car, 4 feet inside
the garage.
a. Determine the velocity of the car when $s(t) = 0$.
b. Determine the velocity of the car when $s(t) = 14$.
156. The position of a hummingbird flying along a straight
line in $t$ seconds is given by $s(t) = 3t^3 – 7t$ meters.
a. Determine the velocity of the bird at $t = 1$ sec.
b. Determine the acceleration of the bird at $t = 1$ sec.
c. Determine the acceleration of the bird when the
velocity equals 0.
157. A potato is launched vertically upward with an initial
velocity of 100 ft/s from a potato gun at the top of an
85-foot-tall building. The distance in feet that the potato
travels from the ground after $t$ seconds is given by
$s(t) = -16t^2 + 100t + 85$.
a. Find the velocity of the potato after 0.5 s and
5.75 s.
b. Find the speed of the potato at 0.5 s and 5.75 s.
c. Determine when the potato reaches its maximum
height.
d. Find the acceleration of the potato at 0.5 s and 1.5
s.
e. Determine how long the potato is in the air.
f. Determine the velocity of the potato upon hitting
the ground.
158. The position function $s(t) = t^3 - 8t$ gives the
position in miles of a freight train where east is the positive
direction and $t$ is measured in hours.
a. Determine the direction the train is traveling when
$s(t) = 0$.
b. Determine the direction the train is traveling when
$a(t) = 0$.
c. Determine the time intervals when the train is
slowing down or speeding up.
159. The following graph shows the position $y = s(t)$ of
an object moving along a straight line.

a. Use the graph of the position function to determine
the time intervals when the velocity is positive,
negative, or zero.
b. Sketch the graph of the velocity function.
c. Use the graph of the velocity function to determine
the time intervals when the acceleration is positive,
negative, or zero.
d. Determine the time intervals when the object is
speeding up or slowing down.
