Chapter 9, Problem 31A

(a)

To determine

To find: The rank of the carts by kinetic energy before they meet the incline.

Answer to Problem 31A

The kinetic energy of the carts is ranked from highest to lowest is $\text{B>A>C>D}$ .

Explanation of Solution

Given:

  ${\text{v}}_1=4\raisebox{1ex}{$\text{m}$}\!\left/ \!\raisebox{-1ex}{$\text{s}$}\right.\text{and}{m}_1=1\text{kg}$

  ${\text{v}}_2=3\raisebox{1ex}{$\text{m}$}\!\left/ \!\raisebox{-1ex}{$\text{s}$}\right.\text{and}{m}_2=2\text{kg}$

  ${\text{v}}_3=2\raisebox{1ex}{$\text{m}$}\!\left/ \!\raisebox{-1ex}{$\text{s}$}\right.\text{and}{m}_3=3\text{kg}$

And,

  ${\text{v}}_4=1\raisebox{1ex}{$\text{m}$}\!\left/ \!\raisebox{-1ex}{$\text{s}$}\right.\text{and}{m}_4=4\text{kg}$

Formula used:

The kinetic energy is

  $KE=\frac{1}{2}m{v}^2$

Where, m is the mass and v is the velocity of the cart.

Calculation:

Now calculate the kinetic energy of the given cart by using the given values such that

  $K{E}_1=\frac{1}{2}m{v}^2=\frac{1}{2}\times 1\text{kg}\times {\left(4\raisebox{1ex}{$\text{m}$}\!\left/ \!\raisebox{-1ex}{$\text{s}$}\right.\right)}^2=8\text{J}$

  $K{E}_2=\frac{1}{2}{m}_2{v}_2{}^2=\frac{1}{2}\times 2\text{kg}\times {\left(3\raisebox{1ex}{$\text{m}$}\!\left/ \!\raisebox{-1ex}{$\text{s}$}\right.\right)}^2=9\text{J}$

  $K{E}_3=\frac{1}{2}{m}_3{v}_3{}^2=\frac{1}{2}\times 3\text{kg}\times {\left(2\raisebox{1ex}{$\text{m}$}\!\left/ \!\raisebox{-1ex}{$\text{s}$}\right.\right)}^2=6\text{J}$

  $K{E}_4=\frac{1}{2}{m}_4{v}_4{}^2=\frac{1}{2}\times 4\text{kg}\times {\left(1\raisebox{1ex}{$\text{m}$}\!\left/ \!\raisebox{-1ex}{$\text{s}$}\right.\right)}^2=2\text{J}$

Conclusion:

Thus, the decreasing order of kinetic energy is

  $\text{B>A>C>D}$

(b)

To determine

To Find: The rank of the carts by how high they go up.

Answer to Problem 31A

The rank of the height from highest to lowest is $\text{A>B>C>D}$ .

Explanation of Solution

Given info.

  ${\text{v}}_1=4\raisebox{1ex}{$\text{m}$}\!\left/ \!\raisebox{-1ex}{$\text{s}$}\right.\text{and}{m}_1=1\text{kg}$

  ${\text{v}}_2=3\raisebox{1ex}{$\text{m}$}\!\left/ \!\raisebox{-1ex}{$\text{s}$}\right.\text{and}{m}_2=2\text{kg}$

  ${\text{v}}_3=2\raisebox{1ex}{$\text{m}$}\!\left/ \!\raisebox{-1ex}{$\text{s}$}\right.\text{and}{m}_3=3\text{kg}$

And,

  ${\text{v}}_4=1\raisebox{1ex}{$\text{m}$}\!\left/ \!\raisebox{-1ex}{$\text{s}$}\right.\text{and}{m}_4=4\text{kg}$

And, the calculated kinetic energies are 8J, 9J, 6J, and 2J.

Formula used:

The height can be calculated by the formula:

  $\begin{array}{l}KE=mgh\\ \Rightarrow h=\frac{KE}{mg}\end{array}$

Here, $m$ is the mass of the object, $g$ is the acceleration due to gravity and $KE$ is the kinetic energy of the object.

Calculation:

The height can be calculated by using the above formula for the respective cases,

  $h_1=\frac{K{E}_1}{m_{1}g}=0.81\text{m}$

  $h_2=\frac{K{E}_2}{m_{2}g}=0.46\text{m}$

  $h_3=\frac{K{E}_3}{m_{3}g}=0.2\text{m}$

  $h_4=\frac{K{E}_4}{m_{4}g}=0.1\text{m}$

Conclusion:

Thus, the decreasing rank is $\text{A>B>C>D}$ .

(c)

To determine

To find: The rank of the carts by potential energy.

Answer to Problem 31A

The rank of the potential energy from highest to lowest is $\text{B>A>C>D}$ .

Explanation of Solution

Given info.

  ${\text{v}}_1=4\raisebox{1ex}{$\text{m}$}\!\left/ \!\raisebox{-1ex}{$\text{s}$}\right.\text{and}{m}_1=1\text{kg}$

  ${\text{v}}_2=3\raisebox{1ex}{$\text{m}$}\!\left/ \!\raisebox{-1ex}{$\text{s}$}\right.\text{and}{m}_2=2\text{kg}$

  ${\text{v}}_3=2\raisebox{1ex}{$\text{m}$}\!\left/ \!\raisebox{-1ex}{$\text{s}$}\right.\text{and}{m}_3=3\text{kg}$

And,

  ${\text{v}}_4=1\raisebox{1ex}{$\text{m}$}\!\left/ \!\raisebox{-1ex}{$\text{s}$}\right.\text{and}{m}_4=4\text{kg}$

And, the calculated kinetic energy are 8J, 9J, 6J, and 2J.

Formula used:

The height can be calculated by the formula,

  $\begin{array}{l}KE=mgh\\ \Rightarrow h=\frac{KE}{mg}\end{array}$

Here, $m$ is the mass of the object, $g$ is the acceleration due to gravity and $KE$ is the kinetic energy of the object.

Calculation:

When the cart reaches the highest point on the incline, then the potential energy at the top is equal to the kinetic energy at the bottom. Therefore, the rank of the potential energy from the highest to lowest is $\text{B>A>C>D}$ .

Conclusion:

Thus, the order is $\text{B>A>C>D}$ .

(d)

To determine

To explain: The difference between the answers for b and c.

Answer to Problem 31A

Answers are different for b and c because the potential energy at the top is equal to the kinetic energy at the bottom.

Explanation of Solution

Given info.

  ${\text{v}}_1=4\raisebox{1ex}{$\text{m}$}\!\left/ \!\raisebox{-1ex}{$\text{s}$}\right.\text{and}{m}_1=1\text{kg}$

  ${\text{v}}_2=3\raisebox{1ex}{$\text{m}$}\!\left/ \!\raisebox{-1ex}{$\text{s}$}\right.\text{and}{m}_2=2\text{kg}$

  ${\text{v}}_3=2\raisebox{1ex}{$\text{m}$}\!\left/ \!\raisebox{-1ex}{$\text{s}$}\right.\text{and}{m}_3=3\text{kg}$

And,

  ${\text{v}}_4=1\raisebox{1ex}{$\text{m}$}\!\left/ \!\raisebox{-1ex}{$\text{s}$}\right.\text{and}{m}_4=4\text{kg}$

And, the calculated kinetic energy are 8J, 9J, 6J, and 2J.

The potential energy at the top is equal to the kinetic energy at the bottom, which is given as,

  $P{E}_{\text{top}}=K{E}_{\text{bottom}}$

Conclusion:

Therefore, the answer is different for both options.