An elastic collision is an idealized collision in which both momentum and kinetic energy are conserved. Real collisions usually lose some kinetic energy to heat, sound, and deformation, so engineers often describe them with a coefficient of restitution.
Collision types
- Perfectly elastic: momentum conserved, kinetic energy conserved
- Inelastic: momentum conserved, kinetic energy decreases
- Perfectly inelastic: bodies stick together after collision
Coefficient of restitution
Along the line of impact (1D case), define
e = (relative speed after)/(relative speed before)
= -(v2 - v1)/(u2 - u1)
Elastic collisions correspond to e = 1. Perfectly inelastic corresponds to e = 0.
1D two-body elastic collision (e = 1)
Using momentum + kinetic energy conservation gives
v1 = ((m1 - m2)/(m1 + m2)) u1 + (2m2/(m1 + m2)) u2
v2 = (2m1/(m1 + m2)) u1 + ((m2 - m1)/(m1 + m2)) u2
Special case: if m1 = m2 and u2 = 0, then v1 = 0 and v2 = u1 (velocity exchange).
Center-of-mass viewpoint
In the center-of-mass frame, an elastic collision reverses the relative velocity direction while keeping its magnitude. This geometric view extends naturally to 2D/3D scattering.
Real-world losses
Even when e is close to 1, some energy is typically lost through material deformation (internal friction), sound, heat, and vibrations. The effective e may depend on impact speed and material properties.
Common pitfalls
- Momentum conservation does not imply kinetic energy conservation.
- Restitution uses relative speed along the impact line, not just individual speeds.