Collision Calculator (Elastic and Inelastic)

What is conserved in a collision?

When two objects collide and no external force acts, total momentum does not change: m₁ × v₁ + m₂ × v₂ = m₁ × v₁' + m₂ × v₂'. Kinetic energy, however, is conserved only in an elastic collision. In every other collision part of it turns into heat, sound and deformation.

Three kinds of collision

• Elastic collision: kinetic energy is conserved. Billiard balls and steel spheres come close.
• Perfectly inelastic collision: the objects stick and move together: v' = (m₁ × v₁ + m₂ × v₂) / (m₁ + m₂). The loss of kinetic energy is as large as it can be.
• General case: the ratio of separation speed to approach speed is the coefficient of restitution: e = (v₂' − v₁') / (v₁ − v₂). e = 1 is elastic and e = 0 is perfectly inelastic.

Example: elastic collision

A 2 kg object moving at 4 m/s hits a 6 kg object at rest elastically. Total momentum is 2 × 4 = 8 kg·m/s. The final velocities are v₁' = (2 − 6) / (2 + 6) × 4 = −2 m/s and v₂' = 2 × 2 / (2 + 6) × 4 = 2 m/s. The lighter object bounces back. Check: 2 × (−2) + 6 × 2 = 8 kg·m/s, and kinetic energy is 16 J before and after.

Example: vehicles that lock together

A 1000 kg car moving at 20 m/s hits a stationary 1500 kg car and the two lock together. The common velocity is 1000 × 20 / 2500 = 8 m/s. Kinetic energy falls from 200 kJ to 80 kJ: about 60 percent has gone into crushing and heat.

Signs

The calculator places object 1 on the left and object 2 on the right. Velocity to the right is positive and to the left negative. For two objects approaching head on, the second velocity is negative.

Limits

The calculation is one dimensional. In oblique collisions (a cut shot in billiards, a crash at a junction) momentum is conserved separately along two axes and this tool is not enough. Spinning objects, friction acting during the impact and external effects such as springs are also outside its scope.