cq1_17_1 solution and discussion

The following is a solution to the given problem.

Please compare the given solution with your solution and submit a self-critique of any error(s) in your solutions, and/or and additional questions or comments you might have.

Simply copy your posted document into a text editor and insert revisions, questions, and/or self-critiques, marking your insertions with ####. Submit using the Submit Work Form.

Excellent student response:

A steel ball rolls down a ramp and collides with a marble, as in the video. The marble is slightly smaller and quite a bit
less massive than the ball.

If they both experience the same force, which will have the greater acceleration?

The marble will have higher acceleration because a=F_net / m and since this marble has the smaller mass, its a (i.e., acceleration) will be greater.

If the forces were exerted during identical time intervals, which would have the greater change in
velocity?

The marble will have the greater change in velocity because it has the greater acceleration, and the two accelerations occur over the same time period.

Instructor Commentary:

ENERGY CHANGE vs. MOMENTUM CHANGE: Often students will answer the second question in terms of kinetic energy. It is true that a greater velocity is associated with a greater kinetic energy, but the relative changes in kinetic energy depend on the initial velocities of the objects and aren't predicatable in a collision.

In this example, it is the time of contact between the two objects that is the same; the objects move through different distances while in contact.

Change in kinetic energy is related to a force acting through a distance (`dKE = F_net_ON * `ds).

When a force acts over a given time interval, it is its momentum that changes in a predictable manner (F_net_ON * `dt = `dp, change in momentum).

The fact that the forces are equal and opposite, and that they act for the same time interval, implies that the momentum changes are equal and opposite.

TECHNICAL NOTES:

Technically one acceleration is positive and the other is negative, and on the number line any positive number is greater than any negative number. Since the positive direction can be chosen arbitrarily we could technically conclude that either object has the greater acceleration.
However the intent of the problem is to figure out which acceleration has the greater magnitude. Technically it should have been stated in these terms, but too many words can confuse the intent, so I've decided to let the present statement stand. Another option would be to use the word 'speed' but that would have complicated the explanation.

The idea is simply that the lesser mass changes speed more quickly. In subsequent problems we will worry about the signs of the changes, but for the purpose of this problem we'll keep it simple.

The following is a solution to the given problem.

Please compare the given solution with your solution and submit a self-critique of any error(s) in your solutions, and/or and additional questions or comments you might have.

Simply copy your posted document into a text editor and insert revisions, questions, and/or self-critiques, marking your insertions with ####. Submit using the Submit Work Form.

The marble will have higher acceleration because a=F_net / m and since this marble has the smaller mass, its a (i.e., acceleration) will be greater.

The marble will have the greater change in velocity because it has the greater acceleration, and the two accelerations occur over the same time period.

Instructor Commentary:

In this example, it is the time of contact between the two objects that is the same; the objects move through different distances while in contact.

Change in kinetic energy is related to a force acting through a distance (`dKE = F_net_ON * `ds).

When a force acts over a given time interval, it is its momentum that changes in a predictable manner (F_net_ON * `dt = `dp, change in momentum).

The fact that the forces are equal and opposite, and that they act for the same time interval, implies that the momentum changes are equal and opposite.

TECHNICAL NOTES:

Technically one acceleration is positive and the other is negative, and on the number line any positive number is greater than any negative number. Since the positive direction can be chosen arbitrarily we could technically conclude that either object has the greater acceleration.

The idea is simply that the lesser mass changes speed more quickly. In subsequent problems we will worry about the signs of the changes, but for the purpose of this problem we'll keep it simple.