If a ball of mass, M, moving at velocity, v, collided with a ball of mass 10M at rest, describe what could happen to the velocities of the balls after the collision. Assume the less massive ball bounces off the more massive ball and moves in the negative direction. Be specific about their relative final velocities compared with the initial velocity of the less massive ball.

Refer to the diagram shown below.

Before collision, the momentum of the two masses is

P₁ = Mv + (10M) * 0 = Mv

After the collision, assume that the lighter ball rebounds off the heavier ball with a coefficient of restitution of r, so that v₂ = rv.

If r = 1, the rebound is elastic and v₂ = - v.

If r < 1, the rebound velocity is v₂ = - rv.

If r = 0, the lighter ball sticks to the heavier ball.

The momentum after collision is

P₂ = - Mv₂ + 10Mv₁

Because momentum is conserved, P₁ = P₂. That is,

10Mv₁ - M (rv) = Mv

v₁ = v (1+r) / 10 for r>0.

When r=1 (elastic rebound)

v₁ = v/5.

The heavier ball moves right at 20% of the velocity of the lighter ball,

and the lighter ball rebounds with its velocity in the opposite direction.

When 0 < r < 1,

v₁ = (1+r) / 10.

The heavier ball travels with greater than 20% of the velocity of the lighter ball, and the lighter ball rebounds with a velocity less than its initial velocity.

When r=0, the balls will stick together and

(10M + M) v₁ = Mv

v₁ = v/11.

The stuck balls move together at 1/11 of the initial velocity of the lighter ball.