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Coefficient of Restitution

The Coefficient of Restitution ($e$) is a dimensionless measure of the elasticity of a collision between two bodies. It is defined as the ratio of the relative speed after collision to the relative speed before collision along the line of impact.

$e=\frac{v_{2f}-v_{1f}}{v_{1i}-v_{2i}}$

Where:

• $v_{1i},v_{2i}$ are initial velocities.
• $v_{1f},v_{2f}$ are final velocities.

Key Properties

• Range: $0\le e\le 1$ for passive collisions.
  • $e=1$: Perfectly elastic collision (kinetic energy conserved).
  • $e=0$: Perfectly inelastic collision (maximum kinetic energy loss; objects stick together).
  • $0<e<1$: Inelastic collision (partial energy loss to heat, sound, deformation).
• Energy Loss: The fraction of kinetic energy lost is proportional to $(1-e^2)$.

Complex Collision Dynamics

While $e$ is often treated as a constant material property, real-world interactions involve complex dependencies on impact velocity, temperature, and object geometry.

• Composite Systems: Combining two highly elastic objects does not guarantee an optimal rebound or simple additive elasticity. Interactions between layers can lead to energy dissipation mechanisms that reduce the effective coefficient of restitution for the system.
• The Zero Bounce Paradox: Specific configurations of similarly elastic objects can result in negligible bounce despite high individual material elasticity, due to phase cancellation or internal damping effects during impact The Paradox of Zero Bounce from Similarly Elastic Objects.

References

• The Paradox of Zero Bounce from Similarly Elastic Objects