Elastic Collision Calculator

Find post-collision velocities for perfectly elastic 1D collisions using momentum + kinetic energy conservation.

🎱 Collisions 📐 Elastic ⚡ KE + p Conserved

Enter masses and initial velocities (positive = right, negative = left). Kinetic energy is fully conserved.

Mass 1 (m₁) kg
Initial velocity 1 (v₁) m/s
Mass 2 (m₂) kg
Initial velocity 2 (v₂) m/s
⚠️ Enter valid positive numbers.

Understanding Elastic Collision

An elastic collision is one in which both momentum and kinetic energy are conserved. No energy is lost to heat, sound, or deformation — the objects bounce off perfectly. Real-world examples include billiard balls, atomic and subatomic particle collisions, and superball bounces. Most macroscopic collisions are inelastic to some degree.

The 1D elastic collision equations are: v₁' = ((m₁−m₂)v₁ + 2m₂v₂)/(m₁+m₂) and v₂' = ((m₂−m₁)v₂ + 2m₁v₁)/(m₁+m₂). These are derived simultaneously from the conservation of momentum (m₁v₁ + m₂v₂ = m₁v₁' + m₂v₂') and kinetic energy (½m₁v₁² + ½m₂v₂² = ½m₁v₁'² + ½m₂v₂'²).

Special cases reveal interesting physics: equal masses (m₁ = m₂) → velocities exchange. A heavy ball hitting a light stationary one: heavy barely slows, light flies off at ~2× the heavy ball's speed. A light ball hitting a massive stationary one: light bounces back at nearly the same speed, massive barely moves.

In physics simulations, particle physics experiments, and game engines, elastic collisions are fundamental building blocks. Thermodynamic temperature in an ideal gas derives from elastic molecular collisions. The coefficient of restitution (e) quantifies elasticity: e = 1 for perfect elastic, 0 for perfectly inelastic.

Formula Reference Table

Solve ForFormulaNotes
Post-collision v₁'v₁' = ((m₁−m₂)v₁ + 2m₂v₂)/(m₁+m₂)Both KE and p conserved
Post-collision v₂'v₂' = ((m₂−m₁)v₂ + 2m₁v₁)/(m₁+m₂)From same two conservation laws
Equal massesv₁' = v₂, v₂' = v₁Velocities exchange
Momentumm₁v₁ + m₂v₂ = m₁v₁' + m₂v₂'Always conserved
Kinetic energy½m₁v₁² + ½m₂v₂² = ½m₁v₁'² + ½m₂v₂'²Only for elastic collision
Coeff. of restitutione = (v₂'−v₁')/(v₁−v₂) = 1Perfect elasticity

3 Worked Examples

Example 1
Equal Mass — Velocity Exchange

Two 2 kg billiard balls: ball 1 at 5 m/s hits stationary ball 2.

  • v₁' = ((2−2)×5 + 2×2×0)/(2+2) = 0 m/s
  • v₂' = ((2−2)×0 + 2×2×5)/(2+2) = 5 m/s
  • Ball 1 stops completely; ball 2 moves at 5 m/s
✓ Ball 1: 0 m/s | Ball 2: 5 m/s (velocities exchanged)
Example 2
Heavy hits Light

3 kg ball at 8 m/s hits stationary 1 kg ball.

  • v₁' = ((3−1)×8 + 0)/(3+1) = 16/4 = 4 m/s
  • v₂' = (0 + 2×3×8)/(3+1) = 48/4 = 12 m/s
  • Heavy slows from 8 to 4; light flies off at 12 m/s
✓ v₁'= 4 m/s | v₂' = 12 m/s
Example 3
Head-On Collision

2 kg at +6 m/s vs 2 kg at −4 m/s (head-on).

  • v₁' = ((2−2)×6 + 2×2×(−4))/(4) = −4 m/s
  • v₂' = ((2−2)×(−4) + 2×2×6)/(4) = +6 m/s
  • Velocities exchange — both reverse!
✓ v₁' = −4 m/s | v₂' = +6 m/s

Real-World Applications

🎱
Billiards & Pool
Pool is essentially applied elastic collision physics. Equal-mass balls exchange velocities — the cue ball stops and the target ball moves at the cue ball's original speed, a direct consequence of m₁ = m₂.
⚛️
Particle Physics
Particle colliders rely on elastic and inelastic collision analysis. Detecting deviations from elastic behavior signals new particles or forces. Rutherford's gold foil experiment (elastic alpha scattering) revealed the atomic nucleus.
🏓
Sports Physics
Table tennis, squash, and hockey pucks all approximate elastic collisions. The coefficient of restitution determines how much KE is retained after each bounce.
💻
Game Physics
Physics engines (Unity, Unreal) implement elastic collisions for rigid body dynamics. The collision equations are solved every frame for realistic ball-bounce and object-interaction simulations.
🌡️
Thermodynamics
Ideal gas molecules undergo elastic collisions with each other and container walls. Temperature is proportional to mean kinetic energy of these collisions; pressure arises from the impulse of molecular impacts on walls.

Common Mistakes to Avoid

⚠️
Forgetting that KE is also conserved

Elastic = momentum AND kinetic energy conserved. Inelastic = only momentum conserved. Many students use only momentum conservation for elastic problems, missing the second equation needed.

⚠️
Using energy alone

One equation (energy) gives two unknowns (v₁', v₂'). You always need both conservation equations simultaneously to solve for both final velocities uniquely.

⚠️
Sign errors in head-on collisions

Velocities are signed. A ball moving left is negative. Head-on collisions require careful sign tracking — both input and output velocities can be negative.

⚠️
Assuming elastic when problem says 'sticks together'

If objects stick together, it is perfectly INELASTIC, not elastic. Always check whether objects bounce separately (elastic/inelastic) or merge (perfectly inelastic).

⚠️
Expecting KE to always be conserved

Only elastic collisions conserve KE. Most real collisions (cars, clay, dropping objects) are inelastic — use only momentum conservation unless specifically told the collision is elastic.

Frequently Asked Questions

What is an elastic collision? ▾
A collision where both total momentum and total kinetic energy are conserved. No energy converts to heat, sound, or deformation. Billiard balls and atomic particle collisions approximate this. In practice, 'elastic' means the coefficient of restitution e ≈ 1.
What is an inelastic collision? ▾
One where kinetic energy is lost (to heat, sound, deformation) but momentum is conserved. Most real collisions are inelastic. 'Perfectly inelastic' means objects stick together — maximum KE loss while conserving momentum.
Why do equal-mass balls exchange velocities? ▾
From the formulas: with m₁ = m₂, v₁' = v₂ and v₂' = v₁. The balls swap velocities. This is the famous Newton's cradle behavior: the first ball stops; the last ball swings out at the same speed.
What is the coefficient of restitution? ▾
e = (relative speed after)/(relative speed before) = (v₂'−v₁')/(v₁−v₂). For elastic: e = 1. For perfectly inelastic: e = 0. Real collisions: 0 < e < 1. A superball has e ≈ 0.9; a clay ball e ≈ 0.1.
Can a moving object reverse direction in elastic collision? ▾
Yes — when a light object hits a much heavier stationary object, the light object bounces back (reverses direction). Formula: if m₁ << m₂ and v₂ = 0, then v₁' ≈ −v₁. The light object rebounds at nearly its original speed in reverse.
How does elastic collision apply to nuclear reactors? ▾
Neutron moderation in reactors uses elastic collisions. Fast neutrons collide with moderator atoms (water, graphite) to lose speed. Maximum energy transfer per collision occurs when masses are equal — hence hydrogen-rich water is an efficient moderator (neutron ≈ proton mass).
Why don't billiard balls have perfect elastic collisions? ▾
Real balls deform slightly at contact, producing sound and heat. High-speed photography shows ball compression during contact. The coefficient of restitution for billiard balls is typically 0.92–0.96, not exactly 1. High-quality balls are denser and harder to approach e = 1.
What is the center of mass frame for collisions? ▾
In the center-of-mass (COM) frame, total momentum = 0. Both objects approach, collide, and recede symmetrically. This frame simplifies elastic collision analysis: each object simply reverses its COM-frame velocity. Converting back to the lab frame gives the final velocities in the original reference frame.

Related Calculators

💢 Inelastic Collision 💨 Momentum 💥 Impulse 🔧 Kinetic Energy ⚡ Newton's 2nd Law

Formula Explorer connections

Interpretation: This relationship connects motion, force, momentum, work or energy in a mechanical system. Assumption: Choose a consistent reference direction and unit system. The model may assume constant acceleration, rigid bodies, negligible losses or an isolated system.

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