9781118230725.pdf

Elastic Collisions in One Dimension

As we discussed in Module 9-6, everyday collisions are inelastic but we can
approximate some of them as being elastic; that is, we can approximate that the
total kinetic energy of the colliding bodies is conserved and is not transferred to
other forms of energy:

. (9-62)

This means:



total kinetic energy

before the collision



total kinetic energy

after the collision

9-7 ELASTIC COLLISIONS IN ONE DIMENSION 237

9-7ELASTIC COLLISIONS IN ONE DIMENSION

After reading this module, you should be able to...

9.32For isolated elastic collisions in one dimension, apply the
conservation laws for both the total energy and the net mo-
mentum of the colliding bodies to relate the initial values to
the values after the collision.

9.33For a projectile hitting a stationary target, identify the re-

sulting motion for the three general cases: equal masses,

target more massive than projectile, projectile more mas-

sive than target.

●An elastic collision is a special type of collision in which

the kinetic energy of a system of colliding bodies is con-

served. If the system is closed and isolated, its linear mo-

mentum is also conserved. For a one-dimensional collision in

which body 2 is a target and body 1 is an incoming projec-

tile, conservation of kinetic energy and linear momentum

yield the following expressions for the velocities immediately

after the collision:

and v 2 f

2 m 1

m 1 m 2

v 1 i.

v 1 f

m 1 m 2

m 1 m 2

v 1 i

Learning Objectives

Key Idea

In an elastic collision, the kinetic energy of each colliding body may change, but

the total kinetic energy of the system does not change.

For example, the collision of a cue ball with an object ball in a game of pool
can be approximated as being an elastic collision. If the collision is head-on
(the cue ball heads directly toward the object ball), the kinetic energy of the cue
ball can be transferred almost entirely to the object ball. (Still, the collision trans-
fers some of the energy to the sound you hear.)

Stationary Target

Figure 9-18 shows two bodies before and after they have a one-dimensional colli-
sion, like a head-on collision between pool balls. A projectile body of mass m 1
and initial velocity v 1 imoves toward a target body of mass m 2 that is initially at
rest (v 2 i0). Let’s assume that this two-body system is closed and isolated. Then
the net linear momentum of the system is conserved, and from Eq. 9-51 we can write
that conservation as
m 1 v 1 im 1 v 1 fm 2 v 2 f (linear momentum). (9-63)

If the collision is also elastic, then the total kinetic energy is conserved and we
can write that conservation as

(kinetic energy). (9-64)

In each of these equations, the subscript iidentifies the initial velocities and the
subscriptfthe final velocities of the bodies. If we know the masses of the bodies
and if we also know v 1 i, the initial velocity of body 1, the only unknown quantities
arev 1 fandv 2 f, the final velocities of the two bodies. With two equations at our dis-
posal, we should be able to find these two unknowns.

1

2 m^1 v^1 i

(^2) ^1
2 m^1 v^1 f
(^2) ^1
2 m^2 v^2 f
2
Figure 9-18Body 1 moves along an xaxis
before having an elastic collision with
body 2, which is initially at rest. Both
bodies move along that axis after the
collision.
x
Before v 1 i
m 1
Projectile
m 2
Target
v 2 i = 0
x
After
v 1 f
m 1 m 2
v 2 f
Here is the generic setup
for an elastic collision with
a stationary target.