9781118230725.pdf

two-body system and one-dimensional collision of Fig. 9-14. From Eq. 9-25
, we can relate to the total linear momentum of that two-body
system by writing

. (9-54)

The total linear momentum is conserved during the collision; so it is given by
either side of Eq. 9-50. Let us use the left side to write

. (9-55)

Substituting this expression for in Eq. 9-54 and solving for give us

. (9-56)

The right side of this equation is a constant, and has that same constant value
before and after the collision.
For example, Fig. 9-16 shows, in a series of freeze-frames, the motion of the
center of mass for the completely inelastic collision of Fig. 9-15. Body 2 is the tar-
get, and its initial linear momentum in Eq. 9-56 is Body 1 is
the projectile, and its initial linear momentum in Eq. 9-56 is Note
that as the series of freeze-frames progresses to and then beyond the collision,
the center of mass moves at a constant velocity to the right. After the
collision, the common final speed Vof the bodies is equal to because then
the center of mass travels with the stuck-together bodies.

:vcom

:p 1 im 1 v: 1 i.

:p 2 im 2 :v 2 i0.

:vcom

:vcom

P

:

m 1 m 2



p: 1 ip: 2 i

m 1 m 2

P :vcom

:

P

:

p: 1 i:p 2 i

P

:

P

:

Mv:com(m 1 m 2 ):vcom

P

:
(P v:com
:
Mv:com)

9-6 MOMENTUM AND KINETIC ENERGY IN COLLISIONS 235

x

m 1

v 1 i v 2 i= 0

m 2

m 1 +m 2

V=vcom

Collision!

vcom

The com of the two

bodies is between

them and moves at a

constant velocity.

Here is the

incoming projectile.

The com moves at the

same velocity even after

the bodies stick together.

Here is the

stationary target.

Figure 9-16Some freeze-frames of the two-body system
in Fig. 9-15, which undergoes a completely inelastic col-
lision. The system’s center of mass is shown in each
freeze-frame. The velocity of the center of mass is
unaffected by the collision. Because the bodies stick
together after the collision, their common velocity
must be equal to :vcom.

V:

:vcom

Checkpoint 7

Body 1 and body 2 are in a completely inelastic one-dimensional collision. What is

their final momentum if their initial momenta are, respectively, (a) 10 kg m/s and 0;

(b) 10 kg m/s and 4 kg m/s; (c) 10 kg m/s and 4kg m/s?