The coefficient of restitution is a key parameter that quantifies the elasticity of a collision between two objects.

During a collision, a portion of the kinetic energy is typically lost due to factors such as deformation and friction. The coefficient of restitution, denoted as $e$, is mathematically defined as the ratio of the relative velocity of separation to the relative velocity of approach:

In this equation, $v_1$ and $v_2$ represent the final velocities of the two objects after the collision, while $u_1$ and $u_2$ denote their initial velocities before the collision. The value of the coefficient of restitution ranges from $0$ to $1$. A value of $0$ indicates a completely inelastic collision, where the objects stick together post-collision, whereas a value of $1$ signifies a perfectly elastic collision, in which no kinetic energy is lost.

The coefficient of restitution serves as a predictive tool for understanding the outcomes of collisions. For instance, if the coefficient of restitution is close to $1$, it suggests that the collision will be elastic, allowing the objects to rebound off each other. Conversely, if the coefficient is near $0$, it indicates that the collision is likely to be inelastic, resulting in the objects adhering to one another after impact.