eloquent logic - momentum - elastic collisions

Legend

p	=	momentum
m	=	mass
v	=	velocity
F_av	=	average force
t	=	time
Vi	=	initial velocity
Vf	=	final velocity

Two pucks with the same mass on an air table. Both the x and y components of momentum are conserved independently.

Elastic Collisions

An elastic collision is an collision where the Kinetic Energy (KE) is conserved. Thus, suppose a head on collision takes place between mass 1 and mass 2 on a horizontal plane with m2 at rest, this completely elastic collision provides:

m₁v_1i + 0 = m₁v_1f + m₂v_2f

and the conservation of KE provides:

1/2 m₁v_1i² + 0 = 1/2 m₁v_1f² + 1/2 m₂v_2f²

If the 2 masses are the same, then we get:

v_1i= v_1f + v_2f
v_1i²= v_1f² + v_2f²

So,

v_1f= v_2f - v_1i

Substitute in v_1i²= v_1f² + v_2f², we get:

v_1i²= (v_2f - v_1i)² + v_2f²

eventually, we get:

2v_2f (v_2f - v_1i) = 0
v_2f - v_1i = 0
v_2f = v_1i

This means that in an completely elastic collision, the KE is completely transferred from mass 1 to mass 2. And mass 1 comes to an compete stop.

Below, m₁ < m₂, and the balls move off in opposite directions.

Below, when m₁ > m₂, and both balls move off in the direction in which m₁ was originally travelling.

Inelastic Collisions

Interactive Lab