Coefficient of Restitution

Parameter that measures the fraction of normal relative speed recovered after impact.

Conceptual context

The coefficient of restitution $e$ is a fundamental dimensionless parameter in classical mechanics that quantifies the elasticity of a collision between two bodies. In the real world, no collision is perfectly elastic; there is always a fraction of kinetic energy that is transformed into other forms of internal energy, such as heat, sound, or permanent plastic deformation of the materials involved. This coefficient acts as a black box that summarizes in a single number the entire complexity of the contact forces, internal shock waves, and molecular dissipations that occur during impact. It is the primary analytical tool for solving collisions without having to delve into the temporal evolution of contact forces, allowing a smooth transition between body kinematics and impact physics.

🟢 Essential level

Central definition

From a phenomenological point of view, the coefficient of restitution $e$ is defined as the ratio between the relative speed of separation after the collision and the relative speed of approach before it. It represents the "elastic memory" of the contact: it is a measure of bounce efficiency, indicating what fraction of the impact velocity survives the encounter to move the bodies apart.

Collision classification

The value of $e$ allows classifying collision types within the ordinary physical spectrum, following a scale of energy restitution.

1. Perfectly elastic collision: Occurs when the parameter is exactly equal to 1. In this ideal limit, there is no kinetic energy loss. The bodies move away with the same relative speed with which they approached. It is the model applicable to collisions between atoms or fundamental particles under specific conditions.
2. Perfectly inelastic collision: Occurs when the parameter is exactly equal to 0. In this case, the bodies stick together after impact and move as a single block. The final relative separation velocity $v_{rel,f}$ is zero. The maximum possible amount of kinetic energy is dissipated compatible with momentum conservation.
3. Inelastic collision: This is the regime that covers the vast majority of macroscopic phenomena, where the value is between the previous limits. Materials partially deform and recover, losing energy in the process.

🔵 Formal level

Basic formulas

Formal definition and kinematic resolution equation

Formally, the coefficient of restitution $e$ is defined as the ratio between the separation and approach relative velocities:

Definition of the coefficient of restitution

This definition provides the necessary equation to close the system of equations for a collision. If we know the masses and the initial velocities $v_{1i}$ and $v_{2i}$, we can predict the final state. The final velocity of the first body $v_{1f}$ can be expressed in terms of the second body's final velocity $v_{2f}$:

Final velocity of body 1 from e

Furthermore, the magnitude known as relative separation velocity $v_{rel,f}$ is obtained directly by applying the coefficient of restitution to the initial approach condition:

Relative separation speed

This relationship is extremely powerful because it allows evaluating the "violence" of the bounce without needing to decompose the velocities into a specific reference system, maintaining the Galilean invariance of the model.

The value of $e$ must always satisfy the physical range constraint:

Ordinary physical range of e

Real student questions

• Can the coefficient of restitution depend on time? The parameter summarizes an instantaneous event. However, if materials heat up after many bounces, their mechanical properties change and the value in the next bounce will be different.
• Is it the same as the friction coefficient? No. Friction opposes tangential motion, while $e$ governs the normal response to the contact plane. They are independent phenomena although both dissipate energy.

Final synthesis

The coefficient of restitution $e$ represents the missing link between the ideal description of collisions and the dissipative reality. By allowing us to model relative velocity loss simply, it becomes an indispensable tool for any dynamic analysis involving contact between solid bodies, providing a clear view of how macroscopic energy inevitably degrades in the physical world.