Coefficient of Restitution (e): Impact Elasticity Ratio

Definition and Relationships

For a one-dimensional collision between bodies 1 and 2, the coefficient of restitution is defined as e = (v_2f − v_1f) / (v_1i − v_2i) where v_i and v_f denote pre- and post-impact velocities along the line of centers. The numerator captures the relative speed after the collision, while the denominator represents the relative speed before impact. Newton’s impact law combines e with conservation of linear momentum to solve for unknown velocities and impulse magnitudes.

When one body is a rigid wall (v₂ = 0), the rebound speed equals e times the incident speed. Drop a ball from height h: it rebounds to h′ = e² h (neglecting air drag). The potential-energy calculator converts between height and energy, illustrating how energy loss scales with e.

Historical and Standardization Notes

Isaac Newton formalized the restitution concept in the Principia (1687), distinguishing “degrees of elasticity” among materials. Modern standards—including ASTM F1887 for baseballs, ASTM C1258 for sports flooring, and ISO 18352 for vehicle crash testing—specify procedures for measuring e under controlled conditions, often defining acceptable ranges for product certification.

Laboratories determine e using drop tests, pendulum impact rigs, or instrumented trolleys. High-speed cameras or accelerometers capture velocities, while force platforms provide impulse data that can be cross-checked against the impulse formulation.

Conceptual Foundations

Energy Considerations

Although e compares velocities, it indirectly measures energy loss. The kinetic-energy ratio after and before collision equals e² when masses are identical. Dissipation stems from plastic deformation, sound, heat, and internal friction. The specific-heat calculator helps estimate temperature rise if the lost energy manifests as heat.

Material and Surface Effects

Material stiffness, damping, and surface roughness influence e. Elastomers approach e ≈ 1 at low strain rates but lose elasticity as temperature increases. Metals exhibit lower e when yielding occurs. Surface condition and lubrication alter contact time, coupling restitution with the coefficient of friction during oblique impacts.

Multi-Dimensional Impacts

In three dimensions, restitution applies along the normal direction, while tangential components involve friction and spin. Sports-ball aerodynamics, for example, require both normal restitution (governing bounce height) and tangential restitution (governing spin retention). Numerical solvers integrate e into constraint-based dynamics to maintain stability in robotics and computer graphics simulations.

Applications and Importance

Sporting regulations specify minimum CoR to ensure consistent gameplay—golf balls, baseball bats, and squash courts all have standards. Automotive engineers model bumper impacts using e values calibrated to crash tests, while packaging designers rely on restitution data to predict how products behave during drops.

Robotics and automation systems program grippers and mobile platforms with target e to control post-contact motion. In planetary science, lander design teams simulate surface interactions using estimated restitution values for regolith, ensuring stability after touchdown.

Measurement Tips and Reporting

Document test conditions: impact velocity, temperature, humidity, contact surface, and number of trials. Because e depends on strain rate, report the velocity range explicitly. Provide uncertainties or repeatability statistics to help downstream analysts incorporate restitution into probabilistic simulations.

When modeling rebounds, convert measured e into post-impact trajectories with the projectile range calculator and cross-check energy budgets with the potential energy tool for consistent unit usage.