Elastic collisions are collisions between objects where both momentum and kinetic energy are conserved. This elastic collision example problem will show how to find the final velocities of two bodies after an elastic collision.

This illustration shows a generic elastic collision between two masses A and B. The variables involved are

m_{A} is the mass of the object A

V_{Ai} is the initial velocity of the object A

V_{Af} is the final velocity of the object A

m_{B} is the mass of the object B

V_{Bi} is the initial velocity of the object B and

V_{Bf} is the final velocity of the object B.

If the initial conditions are known, the total momentum of the system can be expressed as

total momentum before collision = total momentum after collision

or

m_{A}V_{Ai} + m_{B}V_{Bi} = m_{A}V_{Af} + m_{B}V_{Bf}

The kinetic energy of the system is

kinetic energy before collision = kinetic energy after collection

½m_{A}V_{Ai}^{2} + ½m_{B}V_{Bi}^{2} = ½m_{A}V_{Af}^{2} + ½m_{B}V_{Bf}^{2}

These two equations can be solved for the final velocities as

and

If you’d like to see how to get to these equations, see Elastic Collision of Two Masses – It Can Be Shown Exercise for a step by step solution.

### Elastic Collision Example Problem

A 10 kg mass traveling 2 m/s meets and collides elastically with a 2 kg mass traveling 4 m/s in the opposite direction. Find the final velocities of both objects.

### Solution

First, visualize the problem. This illustration shows what we know of the conditions.

The second step is to set your reference. Velocity is a vector quantity and we need to distinguish the direction of the velocity vectors. I’m going to choose from left to right as the “positive” direction. Any velocity moving from right to left will then contain a negative value.

Next, identify the known variables. We know the following:

m_{A} = 10 kg

V_{Ai} 2 m/s

m_{B} = 2 kg

V_{Bi} = -4 m/s. The negative sign is because the velocity is in the negative direction.

Now we need to find V_{Af} and V_{Bf}. Use the equations from above. Let’s start with V_{Af}.

Plug in our known values.

V_{Af} = 0 m/s

The final velocity of the larger mass is zero. The collision completely stopped this mass.

Now for V_{Bf}

Plug in our known values

V_{Bf} = 6 m/s

### Answer

The second, smaller mass shoots off to the right (positive sign on the answer) at 6 m/s while the first, larger mass is stopped dead in space by the elastic collision.

Note: If you chose your frame of reference in the opposite direction in the second step, your final answer will be V_{Af} = 0 m/s and V_{Bf} = -6 m/s. The collision does not change, only the signs on your answers. Make sure the velocity values you use in your formulas match your frame of reference.