Exercise 1

Two carts, A and B, are initially at rest on a horizontal frictionless table as shown in the diagram below. A constant force of magnitude $F_0$ is exerted on each cart as it travels between the 2 marks on the table. Cart B has a greater mass than cart A.

In the box below, enter which cart takes longer to travel between the 2 marks and explain your reasoning.

Use Newton’s second law and the definition of acceleration to derive an equation for each cart relating the net force on the cart to the change in velocity of the cart ($\Delta v_A$ or $\Delta v_B$) and the time interval ($\Delta t_A$ or $\Delta t_B$) that the cart spends between the 2 marks.

Is the quantity $m_A \Delta v_A$ greater, less than, or equal to $m_B \Delta v_B$?

For a constant net force, the quantity $\overrightarrow{F}_\textrm{net}\Delta t$ is called the impulse imparted to the object.

Is the magnitude of the impulse imparted to cart A greater than, less than, or equal to the magnitude of the impulse imparted to cart B? Explain your reasoning.

On the sheet of paper in front of you, derive that the impulse imparted to cart A is equal to the change in momentum of cart A, $\Delta \overrightarrow{p}_A$. This is known as the impulse-momentum theorem.

How does the net work done on cart A ($W_\textrm{net,A}$) compare to the net work done on cart B ($W_\textrm{net,B}$)? Explain.

Is the kinetic energy of cart A greater than, less than, or equal to the kinetic energy of cart B after they have passed the second mark? Explain.

Exercise 2

Consider a ball that rolls from a platform, down a ramp, and then onto a horizontal region as shown in the diagram below.

Suppose ball 1 rolls in the initial direction shown below.

On a sheet of paper, sketch the diagram above and the trajectory that the ball would take. Just do a top down sketch like this:

On your sketch, draw arrows to show the directions of:

1. the acceleration of ball 1 while it is rolling on the ramp.

2. the net force on ball 1 while it is rolling on the ramp.

Suppose ball 2 rolls in the initial direction shown below, starting with the same speed as ball 1.

On a sheet of paper, sketch the diagram above and the trajectory that ball 2 would take. Just do a top down sketch like this:

On your sketch, draw arrows (using the same scale as you did for ball 1) to show the directions of:

1. the acceleration of ball 2 while it is rolling down the ramp.

2. the net force on ball 2 while it is rolling down the ramp.

Compare the net force on ball 1 with the net force on ball 2.

Compare the acceleration of ball 1 with the acceleration of ball 2.

Compare the change in kinetic energy of ball 1 with the change in kinetic energy of ball 2.

If your answer is not consistent with the work done on the ball while on the ramp, revise it. (Also consider the change in potential energy of the ball.)

Compare the final speed of ball 1 with the final speed of ball 2.

For ball 1, on a separate sheet of paper, draw vectors that represent the momentum of the ball at the top of the ramp and at the bottom of the ramp. Use these vectors to construct the change in momentum vector $\Delta \overrightarrow{p}_1$.

If your $\Delta \overrightarrow{p}_1$ vector is not consistent with the impulse-momentum theorem for ball 1, revise it. (Think about the direction of the $\overrightarrow{F} \Delta t$ vector.)

For ball 2, on a separate sheet of paper, draw vectors that represent the momentum of the ball at the top of the ramp and at the bottom of the ramp, using the same scale that you used for ball 1. Use these vectors to construct the change in momentum vector $\Delta \overrightarrow{p}_2$.

If your $\Delta \overrightarrow{p}_2$ vector is not consistent with the impulse-momentum theorem for ball 2, revise it. (Think about the direction of the $\overrightarrow{F} \Delta t$ vector.)

Consider the change in momentum vectors you constructed for ball 1 and for ball 2. How do they compare in direction and magnitude?

Compare the time that ball 1 spends on the ramp with the time that ball 2 spends on the ramp.

If your answer is not consistent with the impulse momentum theorem and change in momentum vectors, revise it.

Exercise 3

A tennis player receives a shot with the ball (0.060 kg) traveling horizontally at 50.0 m/s and returns the shot with the ball traveling horizontally at 40.0 m/s in the opposite direction.

What is the impulse delivered to the ball by the tennis racquet?

What work does the racquet do on the ball?

Exercise 4

In a car crash test, a car of mass 1,500 kg collides with a wall. The initial velocity of the car is 15 m/s in the negative x direction. The final velocity of the car is 2.60 m/s n the positive x direction. The collision lasts for 0.150 s.

Draw a sketch representing this problem.

What is the average force exerted on the car?

Suppose the car did not rebound from the wall, but instead came to rest upon hitting the wall, and did so in the same time interval. Would you expect the average force to be larger, smaller, or the same? Calculate the average force and put it in the box below.