Elastic And Inelastic Collisions In Physics

A collision is a short, strong interaction between two bodies — a car crash, a bat hitting a ball, or two billiard balls clicking together. Collisions are classified as elastic when kinetic energy is conserved along with momentum, and inelastic when some kinetic energy is lost to heat, sound, or permanent deformation. In every collision with no outside forces, total momentum is always conserved; only kinetic energy may or may not be.

This distinction matters because it decides how much damage a crash does and how much energy a game transfers. Car crumple zones are deliberately inelastic — they sacrifice the car to absorb kinetic energy that would otherwise reach the passengers. A perfectly elastic tennis racket, on the other hand, returns nearly all the ball's kinetic energy for maximum bounce. Physicists, engineers, and accident investigators all use these two categories to predict what happens after impact.

Key Concepts

• Momentum (p = mv): mass times velocity, in kg·m/s; always conserved in a collision without external impulses.
• Kinetic energy (KE = ½mv²): conserved only in elastic collisions.
• Elastic collision: both momentum and kinetic energy are conserved; objects bounce apart with no permanent damage or heat.
• Inelastic collision: momentum is conserved but kinetic energy is not — some becomes heat, sound, or deformation.
• Perfectly (completely) inelastic collision: the bodies stick together after impact and move with a common velocity; maximum kinetic energy is lost.
• Impulse: the large force acting over the very short collision time; internal impulses cancel in pairs, which is why total momentum is conserved.

Formulas

• Momentum conservation: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂ (u = before, v = after).
• Elastic collision, one-dimensional: also ½m₁u₁² + ½m₂u₂² = ½m₁v₁² + ½m₂v₂².
• Perfectly inelastic (stick together): common velocity V = (m₁u₁ + m₂u₂) / (m₁ + m₂).
• Head-on elastic, equal masses, one initially at rest: the moving body stops dead and the other takes its full velocity.

Assumptions: external forces (friction, gravity impulses) are negligible during the brief impact, and motion is along one line for the simple formulas. Limits: perfectly elastic collisions are an idealisation — real collisions always lose a little energy; also, relativistic speeds need different momentum formulas.

Worked Example

A 2 kg trolley moving at 5 m/s hits a stationary 3 kg trolley and they stick together. Find their common velocity and the kinetic energy lost.

1. Momentum before: 2 × 5 + 3 × 0 = 10 kg·m/s.
2. Common velocity: V = 10 / (2 + 3) = 2 m/s.
3. KE before: ½ × 2 × 5² = 25 J.
4. KE after: ½ × 5 × 2² = 10 J.
5. Energy lost to deformation and heat: 25 − 10 = 15 J, while momentum stayed at 10 kg·m/s throughout.

Common Misconceptions

• "Momentum is lost in inelastic collisions." Never — momentum is conserved in all collisions without external impulse. It is kinetic energy that is lost in inelastic impacts.
• "Elastic means the objects bounce back to where they came from." Elastic only means kinetic energy is conserved. The directions after impact depend on the masses and velocities — equal masses in a head-on hit simply swap velocities.
• "Heavier objects always win collisions." Outcome depends on both mass and velocity (momentum). A fast light object can easily reverse a slow heavy one if its momentum is larger.

Real-World Applications

• Car safety: crumple zones make crashes deliberately inelastic, converting kinetic energy into bent metal instead of passenger injury.
• Sports: billiards and Newton's cradle are near-elastic; boxing gloves make punches more inelastic to protect hands and heads.
• Ballistic testing: a ballistic pendulum catches a bullet (perfectly inelastic) and the swing height reveals the bullet's speed.
• Space docking: spacecraft rendezvous are gentle, nearly perfectly inelastic manoeuvres — momentum conservation governs the combined motion.

Elastic collisions conserve both momentum and kinetic energy; inelastic collisions conserve momentum but lose kinetic energy to heat, sound, and deformation, with the sticking-together case losing the most. The single equation m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂ holds for every collision — it is the one rule you can always trust at impact.