When two or more objects collide, only internal forces act between them and change their individual momenta and energies. However, in the process of a collision no external force is relevant. Therefore the total momentum is conserved (unchanged) during a collision:

($\sum$_{i=1,...,n i})_initial = ( $\sum$_{i=1,...,n i}) _final

This relationship means that the the (vector) sum of all momenta of all of objects right before the collision is equal to the sum after.

We distinguish between two different limiting cases for which we can perform calculations:

Totally elastic collisions:

By definition, in elastic collisions the total kinetic energy is conserved:

 

( $\sum$_i=1,...,n K_i)_initial = ($\sum$_i=1,...,n K_i)_final

Totally inelastic collisions:

This is the other extreme, where the maximum possible amount of kinetic energy is converted into other energy forms, such as deformation. In this limit, the collision partners all stick to each other after the collision.

 

(v₁)_final = (v₂)_final = ... = (v_n)_final

Inelastic but not totally inelastic collisions:

This is all the cases between the two extreme cases above. Momentum is conserved but energy is lost. However the energy lost is smaller than it would be if the objects stick together after the collision. To solve this use the law of momentum conservation, ($\sum$_{i=1,...,n i})_initial = ($\sum$_{i=1,...,n i}) _final

Then you can find the energy lost with

E_lost = ($\sum$_i=1,...,n K_i)_initial- ($\sum$_i=1,...,n K_i)_final

The importance of these conservation laws for our ability to calculate the final momenta and/or velocities of objects after collisions cannot be overemphasized. In the following, we will introduce several scenarios for which this statement will hold true.

© MultiMedia Physics, 1999