How to Calculate a Collision and the Coefficient of Restitution

Finds the velocities of two bodies after a head-on collision, by combining conservation of momentum with the definition of the coefficient of restitution. A value of 1 is a perfectly elastic collision, and 0 means the two move off together.

This finds the velocities of two bodies after a head-on collision. Two equations are needed. The first is conservation of momentum: $m_1v_1 + m_2v_2$ is the same before and after.

m_1v_1 + m_2v_2 = m_1v_1' + m_2v_2'

The second is the definition of the coefficient of restitution $e$ , the speed at which they separate over the speed at which they approached.

e = -\dfrac{v_1' - v_2'}{v_1 - v_2}

Solving the two together fixes both velocities after the collision.

What the coefficient means

At $e = 1$ the collision is perfectly elastic and no kinetic energy is lost at all. At $e = 0$ the two move off together at the same velocity, and the loss is as large as it can be. In between, part of the energy goes into heat and sound.

Example

A 2 kg body at 5 m/s meets a 3 kg body at $-1$ m/s head on, with a coefficient of 0.5. Afterwards they move at $-0.4$ m/s and 2.6 m/s. The momentum is 7 kg·m/s before and after, while the kinetic energy falls from 26.5 J to 10.3 J, a loss of 16.2 J.

Notes

Velocity carries a direction. Taking rightwards as positive, a body moving left is entered as a negative number.

Momentum is conserved in every collision, but kinetic energy only when $e = 1$ .