3.1 Elastic collisions with a stationary target
We begin with an example of a one-dimensional elastic collision between two particles of identical mass, one of which is initially stationary. Our aim now is to find the final velocity of each particle after the collision.
A particle of mass moves along the -axis with velocity and collides elastically with an identical particle at rest. What are the velocities of the two particles after the collision?
Let the final velocities be and . Conservation of momentum along the -axis gives
and conservation of kinetic energy for this elastic collision gives
By eliminating common factors, Equation 3 can be simplified to give
and Equation 4 can be treated similarly to give
Rearranging Equation 6 gives
the right-hand side of which may be rewritten using the general identity , thus
Dividing both sides of this last equation by ,
and using Equation 5 to simplify the resulting right-hand side gives
Comparing this expression for with that in Equation 5 shows that
The result of Example 1 will be familiar to anyone who has seen the head-on collision of two bowls on a bowling green. The moving one stops, and the one that was initially stationary moves off with the original velocity of the first. In effect, the bowls exchange velocities.
Predict qualitatively (i.e. without calculation) what would happen when a body of mass collides with another body of mass that is initially at rest if:
(a) (The symbol should be read as ‘is very much greater than’.)
Experience should tell you that a high-mass projectile fired at a low-mass target would be essentially unaffected by the collision.
A low-mass projectile fired at a massive target would bounce back with unchanged speed.