Case Studies: Impulse and Force - help2

During a collision, an object experiences an impulse that changes its momentum. The impulse is equal to the momentum change. Knowing that impulse is the product of Force•∆Time and that momentum change is the product of Mass•∆Velocity, one can use the Force•∆Time = Mass•∆Velocity relationship as a guide to thinking about how alterations in m, ∆t, and ∆v affect the force in a collision.

In this question, you will have to compare two collisions of a ball with a catcher's mitt. In one case, the catcher's arm is stiff and there is little backward movement of the mitt and ball once contact begins. In the other case, the catcher's arm is relaxed and the ball and mitt move backwards a bit once the contact begins. Here's how to think about the physics of these collisions:
 

The Variable

First you must determine what the variable is. It is either the velocity change (Delta V), the collision or contact time, or the mass of the baseballs. The two balls have the same mass; so rule out mass. And both baseballs change their velocity from 40 m/s to 0 m/s. So they have the same velocity change. So by careful reading and the process of elimination, the variable in these collisions is the time. When a catcher has a stiff or rigid arm, the ball stops immediately. This leads to the smaller collision time. But if the catcher has a relaxed arm and retracts backwards with the ball during the collision, the time is longer. The relaxed arm collision has the greatest time.
 

Momentum Change and Impulse

You will also have to compare the momentum change and the impulse encountered by these two baseballs. The momentum change is your starting point. Momentum change is the mass multiplied by the velocity change. You have just determined that the mass and the velocity change is the same for both Cases. And so there is no difference in momentum change for these two cases. The momentum change is the same regardless of whether the catcher's arm is rigid or relaxed.

In any collision, the momentum change is equal to the impulse. So if the two Cases have the same momentum change, they will also have the same impulse.
 

Force

Finally, you will have to use F•∆t = m•∆v to compare the Force experienced by the baseballs in the two collisions. So the force is the momentum change divided by the collision time ... that is, m•∆v/∆t. The numerator in this expression is the momentum change (m•∆v). You have just determined that it is the same for both Cases. You also have determined that the collision times (∆t) is greater for the catch made with the relaxed arm. So catching a baseball with a relaxed arm and a retracting motion leads to a smaller force ... due to the greater collision time. And this is why catchers catch the pitched ball with a relaxed arm. The stiff arm catch leads to a greater force ... and that hurts.