9781118230725.pdf

want the average force Favgon the wall during the bombardment — that is, the av-

erage force during a large number of collisions.

In Fig. 9-10, a steady stream of projectile bodies, with identical mass mand

linear momenta mv:,moves along an xaxis and collides with a target body that is

228 CHAPTER 9 CENTER OF MASS AND LINEAR MOMENTUM

Figure 9-10A steady stream of projectiles,
with identical linear momenta, collides
with a target, which is fixed in place. The
average force Favgon the target is to the
right and has a magnitude that depends on
the rate at which the projectiles collide
with the target or, equivalently, the rate at
which mass collides with the target.

Target x

v

Projectiles

fixed in place. Let nbe the number of projectiles that collide in a time interval t.

Because the motion is along only the xaxis, we can use the components of the

momenta along that axis. Thus, each projectile has initial momentum mvand

undergoes a change pin linear momentum because of the collision. The total

change in linear momentum for nprojectiles during interval tisnp.The

resulting impulse on the target during tis along the xaxis and has the same

magnitude of npbut is in the opposite direction. We can write this relation in

component form as

Jnp, (9-36)

where the minus sign indicates that Jandphave opposite directions.

Average Force.By rearranging Eq. 9-35 and substituting Eq. 9-36, we find

the average force Favgacting on the target during the collisions:

(9-37)

This equation gives us Favgin terms of n/t, the rate at which the projectiles

collide with the target, and v, the change in the velocity of those projectiles.

Velocity Change.If the projectiles stop upon impact, then in Eq. 9-37 we can

substitute, forv,

vvfvi 0 vv, (9-38)

wherevi(v) and vf(0) are the velocities before and after the collision,

respectively. If, instead, the projectiles bounce (rebound) directly backward from

the target with no change in speed, then vfvand we can substitute

vvfvivv 2 v. (9-39)

In time interval t, an amount of mass mnmcollides with the target.

With this result, we can rewrite Eq. 9-37 as

(9-40)

This equation gives the average force Favgin terms of m/t, the rate at which

mass collides with the target. Here again we can substitute for vfrom Eq. 9-38

or 9-39 depending on what the projectiles do.

Favg

m

t

v.

Favg

J

t



n

t

p

n

t

mv.

J

:

Checkpoint 5

The figure shows an overhead view of a ball bouncing from a vertical wall without any

change in its speed. Consider the change in the ball’s linear momentum. (a) Is px

positive, negative, or zero? (b) Is pypositive, negative, or zero? (c) What is the direc-

tion of p:?

:p

θ θ

y

x