Question

A ball of mass m moving at a speed v makes a head-on collision with an identical ball at rest. The kinetic energy of the balls after the collision is three fourths of the original. Find the coefficient of restitution.

Answer 1

Hint: We are given the question that both the balls are identical. This implies that they possess the same value of mass. We are given a condition regarding the kinetic energy of the balls as well. From these conditions that are specified, the coefficient of restitution can be found.

Complete step by step answer:
In our question it is said that a ball of mass m moving at a speed v makes a head-on collision with an identical ball at rest and the kinetic energy of the balls after the collision is three fourths of the original. We have to find the value of the coefficient of restitution from these conditions.
Coefficient of restitution is the ratio of the final velocity and the initial velocity which is considered after collision. Usually its values range from 0 and 1.
When we consider the masses to be m, and the final velocities as V1 and V2. Let e be its restitution.
Since final kinetic energy is 3/4th the initial kinetic energy,
\[\begin{align}
  & \dfrac{1}{2}m{{v}_{1}}^{2}+\dfrac{1}{2}m{{v}_{2}}^{2}=\dfrac{3}{4}\times (\dfrac{1}{2}m{{v}^{2}}) \\
 & \Rightarrow {{v}_{1}}^{2}+{{v}_{2}}^{2}=\dfrac{3}{4}{{v}^{2}} \\
 & \Rightarrow \dfrac{[{{({{v}_{1}}+{{v}_{2}})}^{2}}+{{({{v}_{1}}-{{v}_{2}})}^{2}}]}{2}=\dfrac{3}{4}{{v}^{2}} \\
 & \Rightarrow \dfrac{(1+{{e}^{2}}){{v}^{2}}}{2}=\dfrac{3}{4}{{v}^{2}} \\
 & \Rightarrow (1+{{e}^{2}})=\dfrac{3}{2} \\
 & \Rightarrow {{e}^{2}}=\dfrac{1}{2} \\
 & \therefore e=\sqrt{\dfrac{1}{2}} \\
\end{align}\]

Thus, the coefficient of restitution is \[\sqrt{\dfrac{1}{2}}\]

Note:
The coefficient of restitution can usually have values from 0 to 1.
It determines whether the collision is elastic or inelastic.
It is zero for a perfectly inelastic collision and one for an elastic collision.
The students must be able to understand the type of collision from the value of the coefficient of restitution.