Elastic Collision Calculator – Final Velocities and Conservation
An elastic collision is the idealised impact in which both momentum and kinetic energy are conserved. It is the first collision every mechanics course meets, and the one that hides the most algebra: momentum conservation alone gives m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂, one equation with two unknowns. The second equation — energy conservation — is quadratic, and solving the pair by hand is where the arithmetic usually goes wrong. This elastic collision calculator does that algebra exactly and shows the conservation bookkeeping that proves the answer is self-consistent.
The Formulas Behind the Calculator
Instead of solving the quadratic directly, the tool uses the linear pair that is equivalent to it: momentum conservation plus the coefficient of restitution e = (v₂ − v₁)/(u₁ − u₂). Solving them together gives closed forms that cover every case at once:
v₁ = [ m₁u₁ + m₂u₂ + m₂·e·(u₂ − u₁) ] / (m₁ + m₂)
v₂ = [ m₁u₁ + m₂u₂ + m₁·e·(u₁ − u₂) ] / (m₁ + m₂)
m₁, m₂ masses (kg)
u₁, u₂ velocities before impact (m/s, signed)
e coefficient of restitution (1 = perfectly elastic)Setting e = 1 reproduces the familiar elastic results v₁ = ((m₁−m₂)u₁ + 2m₂u₂)/(m₁+m₂). Setting e = 0 collapses both bodies onto the centre-of-mass velocity, the perfectly inelastic case. Everything between the two is a real-world partially elastic bounce, which is why the calculator exposes e as a slider rather than burying it.
What an Elastic Collision Really Does
The cleanest statement of elasticity is that it reverses the relative velocity: v₂ − v₁ = u₁ − u₂. The speed of separation equals the speed of approach. Three familiar results follow immediately:
- Equal masses exchange velocities.A cue ball hitting a stationary object ball stops dead and hands over its full speed — the stop shot, and the reason only the end sphere of a Newton's cradle swings out.
- Light off heavy bounces back. A ball striking a much heavier stationary body rebounds at nearly its incoming speed and transfers almost no energy. In the limit of an infinite target mass this becomes the wall bounce,
v₁ = −u₁. - Heavy through light barely slows. The lighter target is kicked away at up to twice the incoming speed while the heavy body continues almost unchanged.
Energy Transfer and the Mass Ratio
For a stationary target the share of kinetic energy handed over is 4m₁m₂/(m₁+m₂)². That curve peaks at 100% when the masses are equal and falls away symmetrically in the logarithm of the mass ratio. It is the reason nuclear reactors use hydrogen or deuterium rather than lead to slow neutrons: a neutron loses almost all its energy in a single hit on a proton of nearly identical mass, about 28% per collision on carbon, and under 2% on lead.
Oblique Collisions and the 90-Degree Rule
Real balls rarely meet dead centre. In the 2-D mode the tool resolves each velocity into a component along the line of centres and one perpendicular to it. Only the normal components collide; the tangential components pass through untouched. For two equal masses with one initially at rest, energy conservation then forces the outgoing paths to be exactly 90° apart — the signature every pool player relies on without knowing the algebra.
The Centre-of-Mass Frame
Viewed from the frame moving at v_cm = (m₁u₁ + m₂u₂)/(m₁ + m₂), the total momentum is zero before and after. Each body simply turns around with its speed unchanged. Because the collision cannot alter v_cm, this frame is usually the fastest route through an exam question, and the calculator reports every quantity in it alongside the lab-frame answer.
e = 0.90–0.95; a tennis ball on concrete is nearer 0.75, and a lump of clay is close to 0. Enter a measured final velocity and the tool back-computes the real restitution for your setup.Reading the Results
Alongside the two final velocities the calculator reports the total momentum and kinetic energy before and after, the reduced mass μ = m₁m₂/(m₁+m₂), the approach and separation speeds, the impulse exchanged, and — when a contact duration is supplied — the average contact force F = J/Δt. The kinetic energy lost follows ΔKE = ½·μ·(1 − e²)·(u₁ − u₂)², which is identically zero for a perfectly elastic collision and maximal for a perfectly inelastic one.
All inputs are converted to SI before the solve and formatted back into your chosen units afterwards, so mixing pounds with miles per hour is safe. Because the formulas are classical, the tool warns as soon as any speed passes 1% of the speed of light and refuses results at or above c.