How to Solve Elastic and Inelastic Collisions (1D Formulas)

Solve 1D elastic and inelastic collisions with momentum conservation, restitution e, and worked examples. Verify final speeds with a free collision calculator.

What a 1D Collision Problem Asks

A one-dimensional collision is two masses interacting along a line. During the short impact, external forces are usually neglected, so linear momentum is conserved: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂. Kinetic energy may or may not be conserved — that is what separates elastic from inelastic cases.

Pick a positive direction (usually right) and keep it for every velocity before and after the collision. Opposing motion uses opposite signs.

If you already have the masses and speeds and only need checked final velocities, open the free collision calculator first, then return here for the formulas and traps.

Three Standard Cases

Perfectly inelastic (stick together)

The masses share one final speed. Conserve momentum only: v = (m₁u₁ + m₂u₂) / (m₁ + m₂). This loses the maximum kinetic energy allowed by momentum conservation. The coefficient of restitution is e = 0.

Elastic (e = 1)

Conserve both momentum and kinetic energy. The closed-form 1D results are:

v₁ = (u₁(m₁ − m₂) + 2 m₂ u₂) / (m₁ + m₂)

v₂ = (u₂(m₂ − m₁) + 2 m₁ u₁) / (m₁ + m₂)

Special case: equal masses exchange velocities. If m₁ = m₂ and one mass starts at rest, the moving mass stops and the other takes its speed.

Partial restitution (0 < e < 1)

Real collisions often bounce with some energy loss. The restitution coefficient e compares relative speed of separation to relative speed of approach. Use the general 1D formulas with e inserted (the calculator’s partial mode).

Units and Signs

Mass → kilograms (kg).

Velocity → metres per second (m/s). Convert km/h by dividing by 3.6.

Momentum → kg·m/s. Kinetic energy → joules (J).

Keep one sign convention for the whole problem.

Trap: treating a sticky collision as elastic conserves too much energy and gives the wrong final speeds.

Worked Examples

Example 1: equal masses, elastic

m₁ = m₂ = 2.0 kg, u₁ = 3.0 m/s, u₂ = 0. Elastic formulas give v₁ = 0 and v₂ = 3.0 m/s. Momentum before and after is 6.0 kg·m/s; KE stays 9.0 J.

Example 2: perfectly inelastic stick

m₁ = 3.0 kg at 4.0 m/s hits m₂ = 1.0 kg at rest and they stick. v = (12 + 0)/4 = 3.0 m/s. KE falls from 24 J to 18 J.

Example 3: check with momentum

Always verify m₁u₁ + m₂u₂ equals m₁v₁ + m₂v₂ within rounding. If it does not, a sign or formula mix-up is likely.

To build p = mv before you tackle two-body collisions, use the momentum calculator.

To compare energy before and after, use the kinetic energy calculator.

Common Mistakes

Assuming KE is conserved when the problem says “stick” or “inelastic”.

Mixing grams with metres per second without converting mass to kilograms.

Dropping signs when the masses approach from opposite directions.

Using 2D or glancing formulas on a strictly head-on 1D problem.

Sanity check: after a sticky collision the common speed lies between the two initial speeds (or at an extreme if one mass is much larger).

Check With the Free Calculator

The collision calculator solves elastic, perfectly inelastic, and partial-restitution cases with visible steps, momentum checks, and KE before/after.

Collision calculator — 1D elastic and inelastic.

Momentum calculator — p = mv and impulse.

Kinetic energy calculator — KE = ½mv².

Velocity calculator — find speeds before colliding.

Learn the three cases, keep signs consistent, then confirm every answer on the calculator.

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