Elastic Collision Calculator
Enter two masses and their initial velocities to find the final velocities after a perfectly elastic 1D collision. A live 3D scene shows the spheres approach, collide and rebound, and charts confirm that momentum and kinetic energy are both conserved.
Reviewed by the ToolNestr Editorial Team — July 2026
Two ideas that trip students up
1. Elastic collisions bounce apart
The blue and red spheres approach, collide, then separate again — the green arrows show each object keeps its own velocity after impact, just changed.
2. Momentum and kinetic energy both match
Two pairs of bars: total momentum before and after (left pair) and total kinetic energy before and after (right pair). In an elastic collision, each pair comes out equal.
Collision graphs
How it works
The core idea in one line: an elastic collision conserves both momentum and kinetic energy — solving the two conservation equations together gives a fixed formula for each final velocity.
v1' = [(m1−m2)u1 + 2m2u2] / (m1+m2)
final velocity of mass 1
v2' = [(m2−m1)u2 + 2m1u1] / (m1+m2)
final velocity of mass 2
KE_before = KE_after
kinetic energy is conserved (elastic only)
Because both conservation laws hold simultaneously, the two unknowns (v1' and v2') are fully determined by the two masses and two initial velocities — no extra information about the collision itself is needed. Special cases are easy to check: equal masses exchange velocities, and a very heavy stationary target barely moves while a light incoming object bounces straight back.
Worked example 1 — equal masses, one stationary
Given: A 2.0 kg ball moving at 6.0 m/s strikes a stationary 2.0 kg ball head-on. Find both final velocities.
Worked example 2 — unequal masses, both moving
Given: m1 = 3.0 kg at u1 = 4.0 m/s meets m2 = 5.0 kg at u2 = −2.0 m/s (moving the opposite way). Find v1' and v2', and verify kinetic energy.
Momentum before: 3(4)+5(−2)=2 kg·m/s; momentum after: 3(−3.5)+5(2.5)=−10.5+12.5=2 kg·m/s ✓ — both conserved.
Elastic vs inelastic vs perfectly inelastic collisions
All three conserve momentum. What changes is kinetic energy and whether the objects stick together.
| Collision type | Momentum | Kinetic energy | After collision |
|---|---|---|---|
| Elastic | Conserved | Conserved | Objects separate, bounce apart |
| Inelastic (general) | Conserved | Some KE lost to heat/sound/deformation | Objects separate, may deform |
| Perfectly inelastic | Conserved | Maximum KE lost (but not necessarily all) | Objects stick together, move as one |
Momentum conservation holds for every closed collision — it is kinetic energy that distinguishes the types.
Where elastic collisions actually matter
🎱 Billiards and pool
A cue ball striking an identical stationary ball nearly stops dead while the target ball shoots off — the equal-mass elastic-collision result, and the reason pool is playable with predictable physics.
⚛️ Gas molecules and particle physics
Ideal-gas kinetic theory assumes molecules collide elastically, which is why pressure and temperature stay stable over time. Rutherford scattering and other particle-collision experiments use the same equations to infer particle masses.
🪐 Newton's cradle
The swinging-ball desk toy is a chain of near-elastic collisions: momentum and kinetic energy pass almost losslessly from the first ball to the last, which is why exactly one ball swings out on the far side.
Common misconceptions
"All collisions conserve kinetic energy."
Only elastic collisions do. Momentum is always conserved in a closed system, but kinetic energy is conserved only in the idealised elastic case — most real-world collisions (car crashes, a dropped ball of clay) lose kinetic energy to heat, sound and deformation.
"Elastic means the objects stick together."
The opposite: in an elastic collision the objects bounce apart and keep separate velocities. It is the perfectly inelastic collision where objects stick together and move with one shared final velocity.
"A bouncier object always means a more elastic collision."
Bounciness (coefficient of restitution) is related but not identical — a very light superball can bounce dramatically without the collision being perfectly elastic, since some energy still converts to sound and vibration.
"If both objects end up moving, the collision must be elastic."
Not necessarily — many inelastic collisions leave both objects moving too, just with less total kinetic energy than before. The defining test is whether KE_before equals KE_after, not whether motion continues.
Formula sources & further reading
The formulas here are standard, traceable to:
- • OpenStax, University Physics Volume 1 — §9.4 "Types of Collisions" (free, peer-reviewed). openstax.org
- • Halliday, Resnick & Walker, Fundamentals of Physics — Chapter 9, Center of Mass and Linear Momentum, elastic collisions in one dimension.
- • Serway & Jewett, Physics for Scientists and Engineers — Chapter 9, Linear Momentum and Collisions.
Formulas assume a 1D (head-on) collision with no external forces. Results are rounded for display.
How to use this calculator
Enter the masses
Type m1 and m2 — any consistent mass unit works since it cancels in the ratio.
Enter the velocities
Type u1 and u2 in m/s; use negative signs for objects approaching from opposite directions.
Check the physics
Read v1' and v2', then confirm momentum and kinetic energy both match before and after.
Related tools
Frequently asked questions
What is an elastic collision?
An elastic collision is one where both momentum and kinetic energy are conserved. The objects bounce off each other without any permanent deformation, heat, or sound loss — an idealisation that real collisions only approach, such as billiard balls or gas molecules.
What is the formula for elastic collision final velocities?
For two masses m1 and m2 with initial velocities u1 and u2 along one line, the final velocities are v1' = ((m1−m2)u1 + 2m2u2)/(m1+m2) and v2' = ((m2−m1)u2 + 2m1u1)/(m1+m2). Both come from solving momentum and kinetic-energy conservation simultaneously.
What happens when two equal masses collide elastically?
If m1 = m2, the formulas simplify to v1' = u2 and v2' = u1 — the objects simply exchange velocities. This is why a moving billiard ball that hits an identical stationary ball stops dead, transferring all its motion to the other ball.
What if a moving object hits a much heavier stationary one?
As m2 → ∞ with u2 = 0, v1' → −u1 (the light object bounces straight back at the same speed) and v2' → 0 (the heavy object barely moves) — like a ball bouncing off a wall.
Is kinetic energy really conserved, or just close?
By definition, a perfectly elastic collision conserves kinetic energy exactly: ½m1u1² + ½m2u2² = ½m1v1'² + ½m2v2'². This calculator checks that equality using your inputs — real collisions (like most sports and vehicle impacts) always lose a small amount to sound, heat and deformation, so they are only approximately elastic at best.