Elastic Collision

Introduction

An elastic collision is one in which two objects collide and bounce apart with no loss of kinetic energy. Along with momentum, kinetic energy is perfectly conserved throughout the event. Elastic collisions represent an idealised case that real-world interactions approximate — notably in billiards, atomic scattering, and gas molecule interactions — and studying them illuminates two of the most powerful conservation laws in all of physics.

The Physics Explained

Every collision conserves momentum. Momentum is a vector quantity — mass times velocity — and the total momentum of an isolated system never changes, regardless of whether the collision is elastic or not. This follows directly from Newton's third law: the force that ball 1 exerts on ball 2 is equal and opposite to the force ball 2 exerts on ball 1, so their momentum changes cancel out.

Elastic collisions go further: they also conserve kinetic energy. This means that whatever kinetic energy the system had before the collision, it still has exactly that amount after, just redistributed between the two objects. No energy is converted to heat, sound, or deformation. In reality, true elasticity is rare — snooker balls are close, and atomic collisions are essentially perfect — but the elastic model is a useful and clean baseline.

Solving the elastic collision equations for the final velocities reveals some striking special cases. If two identical balls collide (equal masses), the moving ball stops dead and the stationary ball moves off at exactly the original speed — you see this in Newton's cradle. If a heavy ball hits a light stationary ball, both end up moving in the same direction; if a light ball hits a heavy one, the light ball bounces back. The mass ratio is everything.

Key Equations

Conservation of momentum: m₁·v₁ + m₂·v₂ = m₁·v₁' + m₂·v₂'

Conservation of kinetic energy: ½m₁v₁² + ½m₂v₂² = ½m₁v₁'² + ½m₂v₂'²

v₁' = ((m₁−m₂)·v₁ + 2·m₂·v₂) / (m₁+m₂) (final velocity of ball 1)

v₂' = ((m₂−m₁)·v₂ + 2·m₁·v₁) / (m₁+m₂) (final velocity of ball 2)

Key Variables

Symbol	Name	Unit	Meaning
m₁	Mass of ball 1	kg	Mass of the first object
m₂	Mass of ball 2	kg	Mass of the second object
v₁	Initial velocity of ball 1	m/s	Speed and direction of ball 1 before collision
v₂	Initial velocity of ball 2	m/s	Speed and direction of ball 2 before collision
v₁'	Final velocity of ball 1	m/s	Speed and direction of ball 1 after collision
v₂'	Final velocity of ball 2	m/s	Speed and direction of ball 2 after collision
p	Momentum	kg·m/s	Mass times velocity; conserved in all collisions
KE	Kinetic energy	J	½mv²; conserved only in elastic collisions

Real-World Examples

Newton's cradle: The classic desk toy demonstrates elastic collisions between steel balls. Raising one ball and releasing it causes exactly one ball to swing out the other side — momentum and energy are simultaneously conserved.
Billiards and snooker: When the cue ball strikes a stationary ball of equal mass head-on, it nearly stops and the target ball rolls forward — an almost perfectly elastic one-dimensional collision.
Particle physics: In particle accelerators, protons collide at near-light speeds. While these are actually inelastic at the quantum level, the elastic scattering cross-section is a key measurement used to study the structure of matter.

How the Simulation Works

Three sliders let you set the mass of ball 1, the mass of ball 2, and the initial speed of ball 1. Ball 2 starts at rest. Pressing Launch fires ball 1 toward ball 2 on a frictionless track. The collision is handled using the exact elastic collision formulas above — no approximations. After the collision, both balls continue at their computed velocities and bounce off the walls, colliding again whenever they meet. Readouts show the momentum and kinetic energy before and after each collision; you will see they remain constant throughout, confirming conservation.