This collision calculator solves one-dimensional collisions between two masses — perfectly elastic (kinetic energy conserved) or perfectly inelastic (objects stick) — and audits the energy both ways.

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Momentum Is Always Conserved; Energy Only Sometimes

That asymmetry organizes every collision problem: momentum’s conservation is law; kinetic energy’s survival is a property of the collision type, bracketed by the elastic and stick-together extremes. The momentum bookkeeping itself lives on our Momentum Calculator. Part of our Physics Calculators collection.

Worked Example (Verified Both Ways)

  • 2 kg at 5 m/s strikes stationary 3 kg (defaults): elastic → v1′ = −1.0 m/s (rebound), v2′ = 4.0 m/s; KE 25.0 J → 25.0 J exactly
  • Same masses, perfectly inelastic: combined v = 10/5 = 2.0 m/s; KE 25 → 10 J (15 J to deformation/heat — the ballistic-pendulum configuration)
  • Equal masses elastic: velocities simply exchange — Newton’s cradle’s party trick and a useful sanity check (set m1 = m2)
  • Signs are directions: velocities toward each other carry opposite signs; the formulas handle the geometry if you do

Real Collisions Live Between the Extremes

  • The coefficient of restitution e interpolates: e = 1 elastic, e = 0 perfectly inelastic; billiards ≈ 0.95, car crashes ≈ 0.1–0.3.
  • 2D collisions conserve momentum per-axis; glancing blows need vector treatment beyond this 1D page.
  • Explosions are collisions run backwards: KE appears (from chemistry/springs) while momentum still balances.
  • Center-of-mass frame trick: in the CM frame, elastic collisions merely reverse each velocity — the elegant way to check answers.

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Frequently Asked Questions

How do you solve an elastic collision?

With momentum and kinetic energy both conserved: v1′ = [(m1−m2)v1 + 2m2v2]/(m1+m2) and v2′ = [2m1v1 + (m2−m1)v2]/(m1+m2). For 2 kg at 5 m/s hitting stationary 3 kg: v1′ = −1, v2′ = 4 m/s, with KE 25 J on both sides of the ledger.

What happens in a perfectly inelastic collision?

The objects stick and share one velocity: v = (m1v1 + m2v2)/(m1 + m2) — 2.0 m/s in the worked example. This outcome loses the maximum possible kinetic energy (here 15 of 25 J) while conserving momentum exactly.

Why is momentum conserved but not kinetic energy?

Momentum conservation follows from Newton’s third law for any interaction without external impulse — no exceptions in classical mechanics. Kinetic energy, by contrast, converts freely into heat, sound and deformation work during the crunch; only perfectly elastic encounters (atoms, superballs nearly) refuse the conversion.

What is the coefficient of restitution?

e = (separation speed) ÷ (approach speed) along the impact line: 1 for elastic, 0 for sticking, real materials between. This calculator solves the two boundary cases that bracket all real 1D collisions.

How do signs work in collision problems?

Pick a positive direction and sign velocities honestly: approaching objects typically carry opposite signs (e.g. +5 and −3). The formulas are sign-correct — most student errors are bookkeeping, not physics.

Is this collision calculator free?

Yes — free, instant, no sign-up.