Chapter 6, Problem 16E

A roller-coaster car with a mass of 900 kg starts at rest from a point 22 m above the ground. At point B, it is 8 m above the ground. [Express your answers in kilojoules (kJ).]

1. a. What is the initial potential energy of the car?
2. b. What is the potential energy at point B?
3. c. If the initial kinetic energy was zero and the work done against friction between the starting point and point B is 30,000 J (30 kJ), what is the kinetic energy of the car at point B?

To determine

The initial potential energy of the car in kilojoules.

Answer to Problem 16E

The initial potential energy of the car is $194\text{kJ}$.

Explanation of Solution

Given info: The mass of the car is $900\text{kg}$ and the position of the car above the ground is at $22\text{m}$.

Write the expression for the gravitational potential energy at a given height.

$PE=mgh$

Here,

$PE$ is the gravitational potential energy

$m$ is the mass

$g$ is the acceleration due to gravity

$h$ is the height

Substitute $900\text{kg}$ for $m$, $9.8{\text{m/s}}^{\text{2}}$ for $g$ and $22\text{m}$ for $h$ to find the initial potential energy of the car.

$\begin{array}{c}PE=\left(900\text{kg}\right)\left(9.8{\text{m/s}}^{\text{2}}\right)\left(22\text{m}\right)\\ =194040\text{J}\\ \text{=}194040\text{J}\times \frac{{10}^{-3}\text{kJ}}{1\text{J}}\\ \approx 194\text{kJ}\end{array}$

Conclusion:

Therefore, the initial potential energy of the car is $194\text{kJ}$.

To determine

The potential energy of the car at point B.

Answer to Problem 16E

The potential energy of the car at point B is $70.6\text{kJ}$.

Explanation of Solution

Given info: The mass of the car is $900\text{kg}$ and the height of the point B is $8\text{m}$.

Write the expression for the gravitational potential energy at a given height.

$PE=mgh$

Substitute $900\text{kg}$ for $m$, $9.8{\text{m/s}}^{\text{2}}$ for $g$ and $8\text{m}$ for $h$ to find the initial potential energy of the car.

$\begin{array}{c}PE=\left(900\text{kg}\right)\left(9.8{\text{m/s}}^{\text{2}}\right)\left(8\text{m}\right)\\ =70560\text{J}\\ \text{=}70560\text{J}\times \frac{{10}^{-3}\text{kJ}}{1\text{J}}\\ \approx 70.6\text{kJ}\end{array}$

Conclusion:

Therefore, the potential energy of the car at point B is $70.6\text{kJ}$.

To determine

The kinetic energy of the car at point B.

Answer to Problem 16E

The kinetic energy of the car at point B is $93.5\text{kJ}$.

Explanation of Solution

Given info: The initial potential energy of the car is $194040\text{J}$, the potential energy of the car at point B is $70560\text{J}$ and the work done against the friction is $30000\text{J}$.

During the motion of the car, the loss in potential energy is converted to the gain in kinetic energy. As well as the work done against the friction causes the dissipation of the kinetic energy.

The gain in kinetic energy during the motion from the initial point A to the final point B, is obtained as,

$K{E}_{\text{A}\to \text{B}}=P{E}_{\text{A}}-P{E}_{\text{B}}$

Here,

$K{E}_{\text{A}\to \text{B}}$ is the kinetic energy gained during the motion from A to B

$P{E}_{\text{A}}$ is the potential energy at the point A

$P{E}_{\text{B}}$ is the potential energy at the point B

Substitute $194040\text{J}$ for $P{E}_{\text{A}}$ and $70560\text{J}$ $P{E}_{\text{B}}$ to find $K{E}_{\text{A}\to \text{B}}$.

$\begin{array}{c}K{E}_{\text{A}\to \text{B}}=194040\text{J}-70560\text{J}\\ =123480\text{J}\end{array}$

The some of the kinetic energy gained is dissipated for the work done against the friction. Thus, the kinetic energy at the point B is given by,

$K{E}_{\text{B}}=K{E}_{\text{A}\to \text{B}}-W$

Here,

$K{E}_{\text{B}}$ is the kinetic energy at point B

$W$ is the work done against the friction

Substitute $123480\text{J}$ for $K{E}_{\text{A}\to \text{B}}$ and $30000\text{J}$ for $W$ to find the kinetic energy at point B.

$\begin{array}{c}K{E}_{\text{B}}=123480\text{J}-30000\text{J}\\ =93480\text{J}\\ =93480\text{J}\times \frac{{10}^{-3}\text{kJ}}{1\text{J}}\\ \approx 93.5\text{kJ}\end{array}$

Conclusion:

Therefore, the kinetic energy of the car at point B is $93.5\text{kJ}$.