## **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.

$$150. \quad s(t) = 2t^3 - 3t^2 - 12t + 8$$

$$151. \quad s(t) = 2t^3 - 15t^2 + 36t - 10t$$

$$152. \quad s(t) = \frac{t}{1+t^2}$$

153. A rocket is fired vertically upward from the ground. The distance *s* in feet that the rocket travels from the  $\text{ground after } t \text{ 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 + 64.5 $ 

- 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.

 $\text{155. The position function } \operatorname{s}(t) = t^2 - 3t - 4 \text{ 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 \text{ is in seconds.} \text{ In this case, } s(t) = 0 \text{ 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.

- $\text{a. } \quad \text{Determine the velocity of the car when } \ s(t) = 0. $
- $\text{b.} \quad \text{Determine the velocity of the car when } s(t) = 14. $

156. The position of a hummingbird flying along a straight  $\text{line in }\ t\text{ seconds is given by }s(t) = 3t^3 - 7t\text{ meters.} $ 

- $\text{a. } \quad \text{Determine the velocity of the bird at } t = 1 \text{ sec.} $
- $\text{b.} \quad \text{Determine the acceleration of the bird at } t = 1 \text{ 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. $ 

- $\text{a. } \begin{array}{c} \text{Find the velocity of the potato after } \ 0.5 \text{ s} \end{array} \text{ and } $ 5.75 s.
- $\text{b.}\quad\text{Find the speed of the potato at 0.5 s and 5.75 s.} $
- c. Determine when the potato reaches its maximum height.
- $\text{d.}\quad \text{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.

 $\text{158.} \quad \text{The position function} \quad s(t) = t^3 - 8t \quad \text{gives} \quad \text{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.

 $\text{159. The following graph shows the position } \mathbf{y} = \mathbf{s}(t) \text{ of} $ an object moving along a straight line.

![](_page_0_Figure_34.jpeg)

- 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.