Elastic Collision Special Cases
Overview
For a perfectly elastic head-on collision: v₁' = (m₁−m₂)/(m₁+m₂) × u₁ and v₂' = 2m₁/(m₁+m₂) × u₁ (assuming u₂ = 0). Special case — EQUAL masses (m₁ = m₂): v₁' = 0, v₂' = u₁. The first object stops completely and the second moves at the original speed. This is seen in Newton's cradle and billiard balls. Special case — very heavy target (m₂ >> m₁): v₁' ≈ −u₁ (ball bounces back), v₂' ≈ 0. Both formulas and these special cases may appear in exams.
The two equations for elastic collisions
For a perfectly elastic collision between m₁ (initial speed u₁) and m₂ (at rest), using conservation of both momentum and KE gives: v₁ = (m₁ − m₂)/(m₁ + m₂) × u₁ and v₂ = 2m₁/(m₁ + m₂) × u₁. Special cases: if m₁ = m₂: v₁ = 0 (stops), v₂ = u₁ (moves at same speed). If m₁ >> m₂: v₁ ≈ u₁ (barely slowed), v₂ ≈ 2u₁. If m₁ << m₂: v₁ ≈ −u₁ (bounces back), v₂ ≈ 0.
- ⚠Using the elastic collision formula when the collision is inelastic
- ⚠Forgetting the formula only works when one mass is initially at rest