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


A particle of mass m_{1 }and velocity v_{1} collides elastically with a particle of mass m_{2}, with initial velocity v_{2}. 
After the collision, m_{1} has velocity v_{1}', and m_{2} has velocity v_{2}'. What are v_{1}' and v_{2}'? 
To find the velocities of the particles after the collision, you can:
Gee, this seems awfully easy  one simple calculation, a couple of additions and subtractions  does it really work, always? Well, yes, it does. Here is a proof.
Actually, the solution for a onedimensional elastic collision is even easier than that! It can be shown (quite easily) that
v_{1_after} = 2v_{cm}  v_{1}, and v_{2_after} = 2v_{cm}  v_{2}
A 2 kg meatball with a velocity of 5 m/s collides headon, elastically, with a 3 kg meatball with a velocity 5 m/s. If the collision is confined to one dimension, what will be their velocities after the collision?
The center of mass of the twomeatball system is easy to find:
The velocity of the 2 kg. meatball is 2v_{cm } v_{1} = 2(1 m/s)  5 m/s = 7 m/s
The velocity of the 3 kg. meatball is 2v_{cm } v_{2 }= 2(1 m/s)  (5 m/s) = 3 m/s
Can it really be that easy? Yes, it can.
If m_{1} is much more massive that m_{2}, then:
Therefore,
v_{1_after} = 2v_{cm}  v_{1} = 2v_{1}  v_{1} = v_{1} (approximately)
The equations predict that the velocity of m_{1} will be essentially unchanged, which corresponds to experience, and:
v_{2_after} = 2v_{cm}  v_{2} = 2v_{1}  v_{2} (approximately)
If m_{2} started at rest (v_{2} = 0), its velocity after the collision will be approximately 2v_{1}.
If the two masses are equal,
Therefore:
In other words, if the masses are equal, the two objects simply exchange velocities in an elastic collision.
If the masses are equal and m_{2} is initially at rest (a special case of part B above):
The velocities after the collision are:
So the two objects exchange velocities  m_{1} stops and m_{2} takes its velocity. Try it on an air track!