Collisions
A short lesson with diagrams, a worked example, a problem to try, and a quick check. Then quiz yourself on 6 practice questions.
In every collision with no net external force, momentum is conserved. Kinetic energy is conserved only in elastic collisions. When objects stick together (perfectly inelastic), the most kinetic energy possible is lost.
1. Types of collisions
2. Useful patterns
Equal masses, elastic, one at rest: they swap velocities. The first stops; the second moves off at v.
Light object hits a heavy one at rest (elastic): it bounces back at nearly its original speed.
In a 1-D elastic collision, the relative speed is the same before and after (it just reverses).
Ballistic pendulum: two stages. Momentum is conserved during the brief collision (energy isn’t), then energy is conserved during the swing (momentum isn’t, because gravity and tension act).
3. Example, then try it
A 2.0 kg cart at 6.0 m/s hits a 1.0 kg cart at rest and they stick. Find their speed and the kinetic energy lost.
(2.0)(6.0) = (3.0)v, so v = 4.0 m/s.
K: ½(2.0)(36) = 36 J before; ½(3.0)(16) = 24 J after. 12 J (one-third) is lost.
A 0.020 kg bullet embeds in a 1.98 kg block hanging from strings, and the block swings up 0.45 m. Find the bullet’s speed. What fraction of its kinetic energy was lost? (g = 10 m/s²)
Swing (energy): ½v² = gh, so the block’s speed just after impact is v = √(2 × 10 × 0.45) = 3.0 m/s.
Impact (momentum): (0.020)vb = (2.0)(3.0), so vb = 300 m/s.
Energy: bullet ½(0.020)(300)² = 900 J; after impact ½(2.0)(3.0)² = 9.0 J.
Lost: 891 of 900 J, about 99%, mostly to heat and deformation.
Answer: 300 m/s; about 99% of the kinetic energy is lost.
4. Common mistakes
A light ball hits a much heavier ball at rest in a head-on elastic collision. What happens to the light ball?
With m1 ≪ m2, v1′ = (m1 − m2)v1/(m1 + m2) ≈ −v1: it bounces back like a ball off a wall.
Ready to practice?
6 questions on Collisions, each with a hint and an explanation for every wrong answer.