Collisions
The idea
When two objects collide, the forces between them are huge, brief, and usually unknowable in detail — yet the total momentum of the pair is the same before and after, because the collision forces are an internal third-law pair that cancel within the system. Conservation of momentum, m₁v₁ + m₂v₂ = m₁v₁ʹ + m₂v₂ʹ, is the master equation for every collision, as long as outside forces are negligible during the brief impact.
Collisions sort into types by what happens to kinetic energy. In an elastic collision kinetic energy is also conserved — think colliding billiard balls or air-track gliders with springy bumpers. In an inelastic collision some kinetic energy converts to heat, sound, and deformation; in the perfectly inelastic extreme the objects stick together and move with one shared velocity. Momentum survives every type; kinetic energy survives only the elastic kind.
The misconception to root out is that momentum conservation implies energy conservation. A head-on crash where two cars crumple and stop dead conserves momentum perfectly (the totals were equal and opposite), yet nearly all the kinetic energy vanished into bent metal and heat. Check the two ledgers separately — they answer different questions.
Worked example
A 1200 kg car traveling at 20 m/s rear-ends a stationary 800 kg car, and the two lock bumpers and slide together. Find their common speed just after impact and the fraction of kinetic energy lost in the collision.
- Total momentum before: only the moving car contributes, so p = 1200 × 20 + 800 × 0 = 24000 kg·m/s in the direction of travel.
- Locked bumpers mean a perfectly inelastic collision: both cars share one final velocity v, so 24000 = (1200 + 800) × v, giving v = 24000/2000 = 12 m/s.
- Tally kinetic energy before: KE = ½ × 1200 × 20² = 240000 J, all in the moving car.
- Tally kinetic energy after: KE = ½ × 2000 × 12² = 144000 J.
- The collision destroyed 240000 − 144000 = 96000 J of kinetic energy — a 40% loss — which went into crumpling metal, sound, and heat, while momentum stayed exactly 24000 kg·m/s.
- Sanity-check the speed: 12 m/s sits between 0 and 20 m/s and is closer to the heavier car's contribution, exactly what a mass-weighted average should do.
Answer. The wreckage moves off at 12 m/s, and 40% of the original kinetic energy (96000 J) is lost to deformation, sound, and heat.
Check your understanding
- Why is momentum conserved in a collision even when enormous forces are involved and kinetic energy is destroyed?
- How can you tell from a problem's wording whether to treat a collision as elastic, inelastic, or perfectly inelastic?
- What would change in the analysis if the 800 kg car had been rolling toward the other car instead of sitting still?
- Where does the lost kinetic energy go in a real crash, and why does that loss not violate conservation of energy?
Build the foundations first
Collisions builds on these concepts. If any feel shaky, start there.