Change in Momentum: Case 1 vs Case 2

What makes the change in momentum the biggest in Case 1?

Identical balls are dropped from the same initial height and bounce back to half the initial height. In case 1 the ball bounces off a cement floor and in case 2 the ball bounces off a piece of stretchy rubber. In which case is the change in momentum of the ball between the instant just before the ball collides with the floor or rubber and the instant just after the ball leaves the floor or rubber the biggest?

Case 1

Case 2

Same in both

Explanation:

The change in momentum of the ball is biggest in Case 1, where the ball bounces off a cement floor. When the ball hits the cement floor, the time of contact between the ball and the floor is extremely short, resulting in a very high force imparted by the floor on the ball.

When a ball collides with a cement floor, the impulse imparted on the ball by the floor is calculated using the formula:

I = ∆F/∆t

As the force during the collision is very high and the time of contact is very short, the impulse will be significant. According to the impulse momentum theorem, the impulse is equal to the change in momentum. Therefore, the change in momentum will be the biggest in Case 1.

This scenario highlights the importance of understanding the relationship between force, time of contact, and change in momentum in collisions. By analyzing different cases of collisions, we can observe how varying factors impact the outcome of the collision and the resultant change in momentum.