A particle of mass m_{1} and velocity v collides elastically (in one dimension) with a stationary particle of mass m_{2}. What are the velocities of m_{1} and m_{2} after the collision?


A particle of mass m_{1 }and velocity v collides elastically with a particle of mass m_{2}, initially at rest. 
After the collision, m_{1} has velocity v_{1}, and m_{2} has velocity v_{2}. What are v_{1} and v_{2}? 
First of all, note that the problem as stated above can be considered the most general case. If both particles have a velocity before the collision, it is always possible (and quick and easy) to switch to a frame of reference in which one of the particles is at rest, then find the velocities after the collision in this new frame of reference, and then switch back to the original reference frame.
Given that, it can be shown that the velocities of the particles after the collision are given by:
Are these solutions reasonable? Let's look at three cases:
Think of a pingpong ball (m_{1}) colliding with a stationary bowling ball (m_{2}),. First, notice that any time m_{1} is less than m_{2}, v_{1} will be negative. Physically, this means that if the "bullet" particle is less massive than the "target" particle the "bullet" particle will bounce back in the direction from which it came. This is certainly what would happen if a pingpong ball hit a bowling ball. In fact, if m_{1 }is very small compared to m_{2}, the equation for v_{1} becomes approximately:
(Formally, .) So the pingpong ball would bounce back with approximately its original speed. This happens. What about the bowling ball? If m_{1} is approximately zero (compared to m_{2}), then
In other words, the equations predict that the bowling ball would hardly move. This happens.
What if an airtrack glider collides elastically with an identical glider initially at rest, or a billiard ball collides elastically with another billiard ball of the same mass (neglecting spin effects)?
The equations predict that the moving particle will come to rest (v_{1} = 0) and the initially stationary particle will "steal" its velocity (v_{2} = v). In effect, the two particles trade velocities. This happens (Try it!).
What happens if a bowling ball collides with an initiallystationary pingpong ball?
Physically, the equations predict that the bowling ball will continue with about the same velocity (hardly even noticing the collision!), and the pingpong ball will fly away with almost twice the bowlingball's velocity. This happens.