When two particles collide, of course we know that momentum is conserved. On top of this, particles also obey Newton's Law of Restitution:
where
is the coefficient of restitution. The coefficient is taken to be a fixed constant for collisions between any two bodies. We can then write down two equations from the law of conservation of momentum and Newton's Law of Restitution and solve them simultaneously fo find the velocities of the two particles after the collision. For example, if
:
From the first diagram, the momentum is
Therefore, from the Law of Conservation of Momentum,
From Newton's Law of Restitution,
We solve the simultaneous equations,
(1)
(2)
(1)+7*(2) gives
+
Sub this value of
into (2) to get
Since
the collision is inelastic and kinetic energy is lost. The initial kinetic energy is
The final kinetic energy is
So
of energy is lost.