Question

Ball 1 collides with another identical ball 2 at rest. As for what value of coefficient of restitution (e) the velocity of the second ball become half times of 1 after collision
(A) $\dfrac{1}{3}$
(B) $\dfrac{1}{2}$
(C) $\dfrac{1}{4}$
(D) $\dfrac{1}{6}$

Answer 1

Hint
The above problem can be solved by using the concept of the collision. The collision may elastic collision or inelastic collision. If the momentum and kinetic energy of the object remain the same before and after collision then the collision is called the elastic collision. If the momentum only remains conserved then the collision is called the inelastic collision. The coefficient of restitution is the same as the ratio of the velocity of separation after collision to velocity of approach before collision.

Complete step by step answer
Let the mass of ball 1 and ball 2 is m. The initial speed of the ball 1 is ${u_1} = v$ .
Given: The initial speed of the ball 2 is ${u_2} = 0$, the velocity of the second ball after collision is ${v_2} = \dfrac{v}{2}$, the velocity of the ball 1 after collision is ${v_1} = v$
The equation to find the coefficient of restitution is given as:
$\Rightarrow e = \dfrac{{{v_1} - {v_2}}}{{{u_1} - {u_2}}}......\left( 1 \right)$
Substitute v for ${u_1}$, 0 for ${u_2}$, v for ${v_1}$ and $\dfrac{v}{2}$ for ${v_2}$ in the equation (1) to find the coefficient of restitution.
$\Rightarrow e = \dfrac{{v - \dfrac{v}{2}}}{{v - 0}}$
$\Rightarrow e = \dfrac{{\dfrac{v}{2}}}{v}$
$\Rightarrow e = \dfrac{1}{2}$
Thus, the coefficient of restitution is $\dfrac{1}{2}$ and the option (B) is the correct answer.

Note
Always remember that the coefficient of restitution is the ratio of the velocity of separation to velocity of approach. The coefficient of restitution is the property of the material of the objects that participates in the collision. The momentum is defined as the effect of the inertia of the moving particle. The kinetic energy of the particle is defined as the energy of a particle due to the variation in the position of the particle in some duration.