BYTETOOLS

Elastic Collision Calculator

Work out the velocities after a head-on collision from conservation of momentum, with a restitution slider from perfectly elastic to perfectly inelastic.

1

e = 1 is a perfectly elastic collision (billiard balls, ideal gas molecules). e = 0 is perfectly inelastic — the objects stick together. A tennis ball on concrete is around 0.75; a lump of clay is near 0.

1 m/s
v₁′ after
5 m/s
v₂′ after
0 J
Kinetic energy lost
0%
Energy lost

Before and after

QuantityBeforeAfterChange
Velocity of object 14 m/s1 m/s-3 m/s
Velocity of object 20 m/s5 m/s5 m/s
Total momentum20 kg·m/s20 kg·m/s0
Total kinetic energy40 J40 J−0 J

Perfectly elastic — kinetic energy is conserved exactly, and the two objects separate at the same relative speed they approached with.

  • Relative approach speed = |v₁ − v₂| = 4 m/s
  • Relative separation speed = |v₁′ − v₂′| = 4 m/s (= e × approach speed)
  • v₁′ = (m₁v₁ + m₂v₂ + m₂·e·(v₂ − v₁)) ÷ (m₁ + m₂) = 1 m/s
  • v₂′ = (m₁v₁ + m₂v₂ + m₁·e·(v₁ − v₂)) ÷ (m₁ + m₂) = 5 m/s
  • Momentum is conserved whatever e is: 20 kg·m/s before, 20 kg·m/s after.
  • Energy lost = ½·(m₁m₂ ÷ (m₁+m₂))·(1 − e²)·(v₁ − v₂)² = 0 J
  • Centre-of-mass velocity = 2.5 m/s — unchanged by the collision, because no outside force acts.

This is a one-dimensional (head-on) collision: everything moves along a single line, so sign carries direction. Momentum is conserved for every value of e because no external force acts during the impact; kinetic energy is conserved only when e = 1. Worked check — a 5 kg mass at 4 m/s striking a stationary 3 kg mass elastically leaves them at 1 m/s and 5 m/s, with momentum 20 kg·m/s and kinetic energy 40 J both unchanged.

What is the Elastic Collision Calculator?

In a one-dimensional elastic collision v₁′ = ((m₁ − m₂)v₁ + 2m₂v₂) ÷ (m₁ + m₂), with the mirrored expression for v₂′. Momentum is always conserved; kinetic energy is conserved only when the collision is perfectly elastic.

  • Final velocities for any pair of masses in a one-dimensional collision
  • Restitution slider covering elastic, partially elastic and perfectly inelastic cases
  • Before and after table for velocities, momentum and kinetic energy
  • Energy lost reported in joules and as a percentage of the original
  • Centre-of-mass velocity and relative approach and separation speeds
  • All computed locally in your browser with nothing uploaded

How to use the Elastic Collision Calculator

  1. 1

    Set the mass unit and the velocity unit for the whole problem.

  2. 2

    Enter the mass and velocity of each object, using a negative velocity for anything moving to the left.

  3. 3

    Drag the coefficient of restitution slider: 1 for perfectly elastic, 0 for perfectly inelastic.

  4. 4

    Read the two final velocities and the energy lost in the result tiles.

  5. 5

    Check the before and after table for momentum and kinetic energy totals, then copy the results.

About the Elastic Collision Calculator

The ByteTools Elastic Collision Calculator takes two masses and their velocities along a line and returns the velocity of each object after they collide. A coefficient of restitution slider covers the whole range in between: e = 1 is perfectly elastic, e = 0 is perfectly inelastic where the two move off together, and values like 0.75 model a tennis ball bouncing on concrete.

A before-and-after table lists each velocity, the total momentum and the total kinetic energy, with the change in each. Momentum comes out identical every time because no outside force acts during the impact, while the kinetic energy row shows exactly how much went into heat, sound and deformation, both in joules and as a percentage of the original.

Everything is calculated in your browser with plain JavaScript, so no data is uploaded and nothing is stored between visits. Results update the moment you drag the slider, which makes it easy to watch how the split of energy changes as a collision becomes less and less elastic.

Frequently asked questions

What is the formula for an elastic collision?

For a head-on elastic collision, v₁′ = ((m₁ − m₂)v₁ + 2m₂v₂) ÷ (m₁ + m₂) and v₂′ swaps the roles of the two objects. Both momentum and kinetic energy are conserved, and the objects separate at the same relative speed at which they approached.

What is the coefficient of restitution?

It is the ratio of the separation speed to the approach speed, written e. A value of 1 means a perfectly elastic bounce with no energy lost, 0 means the objects stick together, and a tennis ball on concrete sits at roughly 0.75.

Is momentum conserved in an inelastic collision?

Yes. Momentum is conserved in every collision because no external force acts during the impact, whatever the value of e. It is kinetic energy that is lost in an inelastic collision, converted into heat, sound and permanent deformation.

What happens when a moving ball hits an identical stationary ball?

In a perfectly elastic head-on collision between equal masses the velocities simply swap: the moving ball stops dead and the stationary one leaves at the original speed. This is the classic Newton's cradle result and a good sanity check for the calculator.

Does this work for glancing collisions?

No — this tool models a one-dimensional head-on collision where everything moves along a single line and the sign of a velocity carries the direction. Two-dimensional glancing impacts need the momentum components resolved separately along each axis.

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