If a particle of mass m hits a stationary particle in an elastic collision, so that kinetic energy is conserved, (momentum is always conserved in collisions) there are two possibilities.
If the collision is head on, both particles move along the same line after the collision.
From conservation of momentum,
(1)
From conservation of energy,
(2)
Square (1) to give
(3)
(3)-(2) gives
or
Of course
otherwise the particles would pass through each other.
The other possibility is that the particles strike a glancing blow.
Treating momentum as a vector, we can construct the triangle below.
Conservation of kinetic energy gives
- This is Pythagoras Theorem - so the angle between
and
is a right angle.