Inelastic collisions and the coefficient of restitution

Perfectly inelastic collision

A collision is perfectly inelastic when the two bodies remain joined together after the impact and move at a common velocity v. Momentum is always conserved, but some of the kinetic energy is dissipated (deformation, heat, sound):

v = (m1u1 + m2u2) / (m1+m2)

The kinetic energy lost is given by:

Delta_Ec = Ec_before - Ec_after >= 0

This loss is the greatest of all possible collisions between m1 and m2.

Coefficient of restitution e

To classify collisions that lie between elastic (e=1) and perfectly inelastic (e=0), we define:

e = (v2 - v1) / (u1 - u2)

where v1 and v2 are the final velocities and u1 and u2 the initial velocities (1D). For a real-world collision, 0 <= e <= 1.

Type of collision	e	Kinetic energy
Elastic	e = 1	conserved
Inelastic (partial)	0 < e < 1	partially lost
Perfectly inelastic	e = 0	maximum loss

Example: calculation of energy lost

A car weighing 1000 kg travelling at 20 m/s collides with a stationary bollard of very large mass, with e = 0.2. In 1D with u2 = 0:

v = e * u1 = 0.2 * 20 = 4 m/s (the car rebounds at 4 m/s in the opposite direction, i.e. v = -4 m/s according to the convention adopted).

Common pitfalls

• Believing that ‘inelastic’ means that p is not conserved: p is always conserved for an isolated system; only Ec may not be conserved.
• Forgetting that e is defined in terms of relative velocities, not absolute velocities.
• Applying 1D formulas to a 2D problem (oblique collision) without decomposing the system along two axes.