Question

Two identical billiard balls are in contact on a smooth table. A third identical ball strikes them symmetrically and comes to rest after impact. The coefficient of restitution is
a. \[\dfrac {2}{3}\]
b. \[\dfrac {1}{3}\]
c. \[\dfrac {1}{6}\]
d. \[\sqrt {\dfrac {3}{2}}\]

Answer 1

Hint: the coefficient of restitution is the ratio of the final velocity of two object after collision to the initial velocity of two objects before collision. The restitution coefficient, denoted by symbol 'e,' it does not have any quantity and have values ranging from 0 to 1.

Complete answer:
The coefficient of restitution is a quantity that represents the attributes of the colliding materials. The coefficient of restitution provides us with the detail about the elasticity of the collision. A fully elastic collision is one wherein there is no loss of total kinetic energy. The highest value of coefficient of restitution for this kind of collision is e = 1. A completely inelastic collision is one in which all of the kinetic energy of the system is dissipated. They have a value of restitution coefficient of e equal to 0. The majority of real-life collisions occur in the middle of these two above mentioned values.

We know that, Restitution coefficient value is equal to the ratio of relative velocity of separation to relative velocity of approach

By preserving momentum throughout the path of impact. The first direction of the $C_1$ ball is the line of impact. U denotes the first ball's beginning velocity, while v is the end velocity of the 2 stationary balls.
$mu=2mv\cos 30$
$\Rightarrow 2mv\sqrt{\dfrac{3}{2}}$
$u=\sqrt{3}v$

Newton's restitution law along the line of effect
$\dfrac{v-0}{u-\cos 30}=-e$
$\dfrac{v}{\dfrac{\sqrt{3}}{2}u}=e$
$e=\dfrac{2}{3}$

Therefore, coefficient of restitution i.e. $e=\dfrac{2}{3}$. Hence option (a) is the correct answer for this given question.

Note:If e = 0, the collision is completely inelastic. If 0