Elastic collision: two-ball velocities

Ball A of mass 2 kg moves at 6 m/s. Ball B of mass 1 kg moves at 3 m/s in the same direction. They collide elastically. Find the velocity of each ball after the collision.

Step-by-step

1. m₁v₁ + m₂v₂ = m₁v₁′ + m₂v₂′
   Momentum is conserved
2. ½m₁v₁² + ½m₂v₂² = ½m₁v₁′² + ½m₂v₂′²
   Kinetic energy is conserved in an elastic collision
3. v₁′ = (m₁−m₂)/(m₁+m₂)·v₁ + 2m₂/(m₁+m₂)·v₂ = 4
   Solve simultaneously (general 1-D elastic collision)
4. v₂′ = 2m₁/(m₁+m₂)·v₁ + (m₂−m₁)/(m₁+m₂)·v₂ = 7
   Same as above

Why it works

In an elastic collision both momentum and kinetic energy are conserved. The two conservation statements together fix the two unknown velocities after the collision.

Common mistakes

• Using momentum conservation alone and leaving one velocity undetermined.
• Treating the elastic collision as perfectly inelastic and giving a single common speed.

Related

inelastic-collision conveyor-belt bullet-embed-block-common-speed