Coefficient of Restitution or Coefficient of Resilience

Typically, real-world collisions between objects don’t conform to the ideal models of completely elastic or inelastic interactions. These encounters are often categorized as imperfect or semi-elastic. The degree to which a collision approximates an elastic encounter is quantified by the coefficient of restitution, a measure of how the relative speed after the collision compares to the relative speed before the collision. This coefficient is denoted as ‘e’.

$$e=\frac{{\text{relative velocity after collision}}}{{\text{relative velocity before collision}}}$$

or

$$\boxed{e=\frac{{{{v}_{2}}-{{v}_{1}}}}{{{{u}_{1}}-{{u}_{2}}}}}$$

Here, \( u_1, u_2 \) represent the velocities of the two bodies before the collision, and \( v_1, v_2 \) their velocities after the collision.

In a perfectly elastic collision where no kinetic energy is lost, relative velocity of separation after collision is equal to relative velocity of approach before collision. Therefore, the value of \( e \) is 1:

$$ \boxed{e = 1} $$

In a perfectly inelastic collision, where bodies coalesce and stop moving relative to each other, \( e \) is 0:

$$ \boxed{ e = 0} $$

In all other types of collisions, the coefficient of restitution falls between 0 and 1:

$$ \bbox[15px, #e4e4e4, border: 2px solid #000000]{\boldsymbol {\color{#000000}{0 < e < 1}}} $$