We want to use the conservation laws for linear momentum to study a fascinating and important type of physical phenomenon: collisions. What first comes to mind when we hear this word is two objects banging into each other, and this is the situation which we shall first consider. However as you will see tomorrow and in problems there are a large range of situations which we would not normally think of as collisions put which are in fact accounted for by similar considerations as work for the standard case of collision between two particles. Basically what we shall understand by a collision is
An event within an isolated system of particles in which internal forces dominate external forces over a specific limited period of timeBecause we have stated that internal forces dominate the dynamics we have that
The total linear momentum is conserved in a collision:
First consider the simplest case of a collision between two particles denoted 1 and 2. We label properties of particles by their number and physical quantities which change in the collision by i and f for initial and final.
Because it is a collision linear momentum of the system consisting of
both particles is conserved. Thus we write
for the momentum of the particles before the collision and
for the total momentum after the collision. We have
Note that as opposed to the energy conservation law momentum
conservation in general produces a vector equation so we really
have as many equations as dimensions in the collision problem. On the other
hand there are also lots of unknowns. Say we know the
initial conditions ie 
, and 
but do not know the final velocities 
and 
.
Momentum conservation produces half the number of equations we need
so more information will be necessary. We get one more equation
from the requirement that energy is conserved
Here I introduced 
which is the increase in the internal
energy of the system following the collision. If I am in one dimension
and I know then I have two equations with two unknown.
As soon as I go to higher dimensions I still do not have enough equations
to figure out what happens after a collision with known initial parameters.
We consider simple examples first:
