CK-12-Physics - Intermediate

7.2. Impulse http://www.ck12.org

Case 1

There are several ways to work the problem. We will find the time it takes the ball to come to rest in each case and
then use the change in momentum to solve for the force.

To find the time we divide the displacement of the ball by its average velocity, that is,

vavg=

( 0 + 46. 95 )

2

= 23. 475 → 23. 48

m

s

x=vavg∆t→∆t=

x

vavg

=

0. 010 m

23. 475 ms

= 4. 26 × 10 −^4 → 4. 3 × 10 −^4 s

Recall that the change in momentum was calculated already as− 6. 81 kgs∗m. We now useF=∆∆pt = −^6.^81

kg∗m
s
4. 26 × 10 −^4 s=
− 15 , 986 →− 1. 60 × 104 N

This is a very large force, equivalent to 3,600 pounds. In all likelihood, this would result in serious damage to the
catcher’s wrist.

Case 2

Since the displacement is ten times greater in this case, the time to bring the ball to rest will be ten times longer
→∆t= 230 ..^10475 mm
s
= 4. 26 × 10 −^3 → 4. 3 × 10 −^3 s

And as beforeF=∆∆tp= −^6.^81

kg∗m

s

4. 26 × 10 −^3 s=−^1 ,^599 →−^1.^60 ×^10

(^3) N. Even though the force in Case 2 is ten times
smaller, you wouldn’t want to do this without a really good catcher’s mitt!
Check Your Understanding
Why is it less painful to fall on a mattress bed than on concrete?
Answer:Let’s say you’re standing up and decide to let yourself fall over like a slightly tipped bowling pin. In one
case, at floor level, there is a mattress and in another case there is concrete. We’ll assume in each case you have
the same initial velocity (close to zero) as you begin your fall. The velocityv 1 with which you hit both the mattress
and the concrete is the same, since you fall through the same displacement. Let’s define velocityv 1 as the initial
velocity of impact and the final velocity as zero, since the mattress and concrete have brought you to rest. In each
case, the change in your velocity is the same as you’re brought to rest, and your mass is the same, as well. Therefore,
the change in your momentum (and therefore impulse) is the same in both cases. The only difference is, since the
mattress “gives” more than the concrete, it will take more time for the mattress to bring you to rest than the concrete.
As we saw above, if the change in momentum is constant, force is decreased the longer it takes to come to rest.