Elastic Collisions & the Kinetic Theory of Gases

Two conservation laws, one predictable outcome

An elastic collision is one in which two bodies bounce apart while conserving both total momentum and total kinetic energy. Momentum conservation alone applies to every collision with no external forces — it's the additional survival of kinetic energy that makes an elastic collision special, and it pins the outcome down completely from the masses and initial velocities. For a one-dimensional head-on collision between masses m₁ and m₂ moving at u₁ and u₂, solving the two conservation equations together gives the post-collision velocities directly.

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂                           (momentum)
½m₁u₁² + ½m₂u₂² = ½m₁v₁² + ½m₂v₂²                    (kinetic energy)

v₁ = ((m₁ − m₂)/(m₁ + m₂))·u₁ + (2m₂/(m₁ + m₂))·u₂

Special case m₁ = m₂:  v₁ = u₂, v₂ = u₁  (velocities simply swap)

live demo · particles colliding and redistributing kinetic energy● LIVE

From billiard balls to gas pressure

The kinetic theory of gases models a gas as an enormous number of tiny molecules colliding elastically with each other and with the container walls, and nothing else. Because every collision conserves kinetic energy, the gas never loses internal energy on its own — it can sustain a steady temperature indefinitely. Pressure emerges directly from this picture: each elastic bounce off a wall reverses a molecule's perpendicular momentum, delivering a tiny impulse, and summing these impulses across astronomically many collisions per second produces the steady pressure a gauge reads. Shrink the container and molecules hit the walls more often — Boyle's law falling straight out of mechanics.

Temperature is average kinetic energy

The kinetic theory ties temperature directly to molecular motion: the average translational kinetic energy of a molecule is proportional to absolute temperature, ½m<v²> = (3/2)k_BT. But molecules don't all move at the same speed — repeated random elastic collisions naturally produce a spread of speeds described by the Maxwell-Boltzmann distribution, with many molecules at moderate speed and a long tail of faster ones. That statistical spread underlies phenomena from evaporation to chemical reaction rates, and it's precisely what a large population of elastically colliding particles converges to over time.

Frequently asked questions

What makes a collision elastic rather than just momentum-conserving?

Momentum is conserved in every collision as long as no external forces act, elastic or not. What distinguishes an elastic collision is that total kinetic energy is also conserved — no energy is converted into heat, sound or permanent deformation, which is what makes the outcome fully predictable from masses and initial velocities alone.

Why do equal-mass particles simply swap velocities in a head-on collision?

Solving the combined momentum and kinetic-energy equations for m1 = m2 makes the final velocity of each particle equal to the other's initial velocity. This is exactly what happens on a pool table when a moving ball strikes a stationary one head-on: the moving ball stops and the struck ball departs at the original speed.

How do elastic collisions between molecules produce gas pressure?

Pressure is the cumulative effect of countless molecules elastically bouncing off the container walls. Each collision reverses a molecule's momentum perpendicular to the wall, delivering a tiny impulse; summed over astronomically many impacts per second, these impulses add up to the steady pressure a gauge reads.

Try it live

Everything above runs in your browser — open Elastic Collisions — 2D Billiards & Maxwell-Boltzmann and watch N frictionless discs bounce while total kinetic energy, momentum and the emerging speed distribution stay in view. Nothing is installed, nothing is uploaded.

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