### An overview

An elastic collision is a collision where both the momentum and the kinetic energy of the objects are conserved. There are two equations to be remembered when analyzing collisions

Kinetic Energy = 1/2mv^2 = 1/2m(v.v)

Momentum = mv

In these equations, m stands for the mass of the object being measured, and v its velocity. In the first equation the "." refers to the dot product of v with itself. This is used when in two or more dimensions, where v is a vector.

### Collision between two objects of equal mass

The most common example of an elastic collision is a collision between two billiard balls, call them ball A and ball B. Ignoring friction, if one was to measure the energy and momentum of the balls before and after they collide one would find that they were equal. Even more specifically, a collision between two billiard balls is an example of an elastic collision between two objects of equal mass. These collisions are interesting because the resulting objects will always move away at right angles from each other. To prove this we simply look at our two equations. Using the principle of relativity, we will center our frame of reference around the start position and velocity of one of the balls, ball B. This allows us to cancel one of the terms in the first half of the equation

1/2m*v_{A1}^2 = 1/2m*v_{A2}^2 + 1/2m*v_{B2}^2

m*v_{A1} = m*v_{A2} + m*v_{B2}

Where m is the mass of a ball, v_{A1} is velocity of A before collision, v_{A2} is velocity of A after collision, etc. By manipulating these equations we can show that

v_{A2} . v_{B2} = 0

From linear algebra, and the definition of the dot product, this means that v_{A2} and v_{B2} are perpendicular. This could mean a couple different outcomes. The trivial solution is that the balls missed each other and v_{B2} remained 0. The next solution is a direct collision where ball A hit ball B directly along its center of mass. This means that v_{A2} is now zero and v_{B2} is equal to v_{A1}. These two are the only solutions possible in one dimension. However in two or more dimensions, there is the possibility that the collision was not straight on, and both will move off with a lesser velocity than the original velocity of ball A. However the dot product guarantees us that their velocities will be at right angles to each other. Think of this the next time you're playing pool.

### Direct collision between two objects of unequal mass

Moving up in complexity, we can consider the case of a direct collision, meaning two objects headed straight at one another along one dimension, between two objects of non-equal mass. We will continue to call these ball A and ball B, but they now have mass m_{A} and m_{B}, respectively. We continue to use the principle of relativity to remove one of the terms from the first half of the equation, and our new equations are

1/2m_{A}*v_{A1}^2 = 1/2m_{A}*v_{A2}^2 + 1/2m_{B}*v_{B2}^2

m_{A}*v_{A1} = m_{A}*v_{A2} + m_{B}*v_{B2}

To simplify matters, we can multiply the equations by 1/m_{A} and substitute u = m_{B}/m_{A}. We can also multiply the energy equation by 2 to cancel out the 1/2's. Doing this grants us

v_{A1}^2 = v_{A2}^2 + u*v_{B2}^2

v_{A1} = v_{A2} + u*v_{B2}

We will ignore the trivial solution where no change occurs and the balls velocities remain the same as at the beginning. We can then rearrange these equations to get both v_{A2} and v_{B2} in terms of u and v_{A1}, which can be measured at the start of the incident. We end up getting

v_{A2} = (2/(u+1) - 1)*v_{A1}

v_{B2} = 2/(u+1)*v_{A1}

If we take a look at Sverre's example, at the start of the incident we are given

v_{A1} = 30

m_{A} = 5

m_{B} = 3

u = m_{B}/m_{A} = 3/5

And plugging these into our equations gives us

v_{A2} = 7.5

v_{B2} = 37.5

As was expected the tossed ball, A, moves forward at a slower speed. The light ball, B, that was hit moves forward at a faster speed than A was originally moving at. If you want to check this simply plug all the variables back into the original two equations, you should get identity.

As the equations are used to describe more and more complex situations, the math can become more obfuscated however the principles remain the same. These two equations based on the conservation of energy and the conservation of momentum can predict everything you would want to know about elastic collisions.