9781118230725.pdf

Inelastic Collisions in One Dimension

One-Dimensional Inelastic Collision

Figure 9-14 shows two bodies just before and just after they have a one-

dimensional collision. The velocities before the collision (subscript i) and after

the collision (subscript f) are indicated. The two bodies form our system, which is

closed and isolated. We can write the law of conservation of linear momentum for

this two-body system as

,

which we can symbolize as

(conservation of linear momentum). (9-50)

Because the motion is one-dimensional, we can drop the overhead arrows for

vectors and use only components along the axis, indicating direction with a sign.

Thus, from pmv, we can rewrite Eq. 9-50 as

m 1 v 1 im 2 v 2 im 1 v 1 fm 2 v 2 f. (9-51)

If we know values for, say, the masses, the initial velocities, and one of the final ve-

locities, we can find the other final velocity with Eq. 9-51.

One-Dimensional Completely Inelastic Collision

Figure 9-15 shows two bodies before and after they have a completely inelastic

collision (meaning they stick together). The body with mass m 2 happens to be ini-

tially at rest (v 2 i0). We can refer to that body as the targetand to the incoming

body as the projectile.After the collision, the stuck-together bodies move with

velocityV.For this situation, we can rewrite Eq. 9-51 as

m 1 v 1 i(m 1 m 2 )V (9-52)

or. (9-53)

If we know values for, say, the masses and the initial velocity v 1 iof the projectile,

we can find the final velocity Vwith Eq. 9-53. Note that Vmust be less than v 1 ibe-

cause the mass ratio m 1 /(m 1 m 2 ) must be less than unity.

Velocity of the Center of Mass

In a closed, isolated system, the velocity of the center of mass of the system

cannot be changed by a collision because, with the system isolated, there is no net

external force to change it. To get an expression for :vcom, let us return to the

:vcom

V

m 1

m 1 m 2

v 1 i

p: 1 ip: 2 ip: 1 fp: 2 f



total momentum P

:

i

before the collision

total momentum P

:

f

after the collision

234 CHAPTER 9 CENTER OF MASS AND LINEAR MOMENTUM

Figure 9-14Bodies 1 and 2 move along an
xaxis, before and after they have an
inelastic collision.

m 1 m 2

Before

Body 1 Body 2

x

v 1 i v 2 i

m 1 m 2

After

x

v 1 f v 2 f

Here is the generic setup

for an inelastic collision.

Figure 9-15A completely inelastic collision between

two bodies. Before the collision, the body with mass

m 2 is at rest and the body with mass m 1 moves

directly toward it. After the collision, the stuck-

together bodies move with the same velocity V.

:

m 1

Projectile

m 2

Target

x

x

V

v 1 i

After

Before

m 1 + m 2

v 2 i = 0

In a completely inelastic

collision, the bodies

stick together.