What is the coefficient of restitution?

Hint: During a collision process between two bodies, there can be a change in the velocity of the two bodies undergoing collision. So the ratio between the difference of the final velocity to the difference of initial velocities of the two bodies is known as the coefficient of restitution.

Complete step by step answer:
Consider two bodies of mass ${{\text{m}}_{\text{1}}}\text{ and }{{\text{m}}_{2}}$ travelling with a velocity ${{\text{u}}_{1}}\text{ and }{{\text{u}}_{2}}$ initially. After the collision the velocity of these two masses changes to ${{\text{v}}_{1}}\text{ and }{{\text{v}}_{2}}$.
Suppose the two masses are undergoing elastic collision in which the kinetic energy and the momentum of the system is conserved. So, we can write,
$\dfrac{1}{2}{{m}_{1}}{{u}_{1}}^{2}+\dfrac{1}{2}{{m}_{2}}{{u}_{2}}^{2}=\dfrac{1}{2}{{m}_{1}}{{v}_{1}}^{2}+\dfrac{1}{2}{{m}_{2}}{{v}_{2}}^{2}$ …. Equation (1)
${{m}_{1}}{{u}_{1}}+{{m}_{2}}{{u}_{2}}={{m}_{1}}{{v}_{1}}+{{m}_{2}}{{v}_{2}}$ ……Equation (2)
From equation (1), we can write.
${{m}_{1}}\left( {{u}_{1}}^{2}-{{v}_{1}}^{2} \right)={{m}_{2}}\left( {{v}_{2}}^{2}-{{u}_{2}}^{2} \right)$
${{m}_{1}}\left( {{u}_{1}}-{{v}_{1}} \right)\left( {{u}_{1}}+{{v}_{1}} \right)={{m}_{2}}\left( {{v}_{2}}-{{u}_{2}} \right)\left( {{v}_{2}}+{{u}_{2}} \right)$…. Equation (3)
From equation (2), we can write
${{m}_{1}}\left( {{u}_{1}}-{{v}_{1}} \right)={{m}_{2}}\left( {{v}_{2}}-{{u}_{2}} \right)$ … Equation (4)
Substituting equation (4) in equation (3), we get
${{u}_{1}}+{{v}_{1}}={{u}_{2}}+{{v}_{2}}$
$\Rightarrow {{u}_{1}}-{{u}_{2}}={{v}_{2}}-{{v}_{1}}$
$e=1=-\dfrac{{{v}_{2}}-{{v}_{1}}}{{{u}_{2}}-{{u}_{1}}}$
Where, e is the coefficient of restitution.
${{v}_{2}}-{{v}_{1}}$ is the relative velocity after collision.
${{u}_{2}}-{{u}_{1}}$ is the relative velocity before collision.
In the case of elastic collision, you can see that the value of e is 1. Which means that the relative velocity after the collision and the relative velocity before collision remains the same.
So the coefficient of restitution can be defined as the ratio of relative velocity after the collision to the relative velocity before the collision.

Note: For an inelastic collision, the coefficient of restitution is always less than one and greater than zero $\left( 0 < e < 1 \right)$. Since some of the energy is lost in the form of heat or sound during the collision, the kinetic energy before the collision will not be equal to the kinetic energy after the collision. But the momentum remains conserved in all types of collision.
If the value of e is less than zero or negative, it means that the two objects continue to move in the same direction before and after the collision.
Coefficient of restitution can also be defined as the ratio of the square root of the final kinetic energy after the collision to the square root of the initial kinetic energy before collision.
$e=\sqrt{\dfrac{\text{Final K}\text{.E}}{\text{Initial K}\text{.E}}}$