Two identical billiard balls are in contact on a table. A third identical ball strikes them symmetrically and come to rest after impact. The coefficient of restitution is

The correct option is 2/3.

Step-by-step Explanation:

Let the three identical billiard balls be $A$ (the striking ball), $B$, and $C$ (the target balls). Let $m$ be the mass of each ball, and $R$ be the radius of each ball.

1. Geometry of the Collision:
Initially, the two identical balls $B$ and $C$ are in contact. When ball $A$ strikes them symmetrically, it is in contact with both $B$ and $C$ simultaneously. At the instant of impact, the distance between the centers of any two touching balls is equal to $2R$.
Therefore, the centers of the three balls form an equilateral triangle of side $2R$.
By symmetry, the line of motion of ball $A$ (let it be the x-axis) bisects the angle between the lines of impact.
The line of impact for ball $B$ is the line joining the centers of $A$ and $B$, which makes an angle of $\theta =30°$ with the line of motion of ball $A$.
Similarly, the line of impact for ball $C$ makes an angle of $-30°$ with the line of motion of ball $A$.

2. Conservation of Linear Momentum:
Let $v_0$ be the initial velocity of ball $A$ along the x-axis.
Since the collision is symmetrical and the surface of the balls is smooth, the impulses transmitted to $B$ and $C$ act along the respective lines of impact. Consequently, after the collision, ball $B$ moves with a velocity $v$ along the line of impact $AB$ (at $30°$ to the x-axis), and ball $C$ moves with the same velocity $v$ along the line of impact $AC$ (at $-30°$ to the x-axis).
We are given that the striking ball $A$ comes to rest after the impact.

Conserving linear momentum along the direction of initial motion (the x-axis):

$m{v}_0=0+mv\cos 30°+mv\cos 30°$

$v_0=2v\cos 30°$

$v_0=2v\left(\frac{\sqrt{3}}{2}\right)=\sqrt{3}v$

$v=\frac{v_0}{\sqrt{3}}$

3. Calculating the Coefficient of Restitution:
The coefficient of restitution ($e$) is defined along the line of impact:

$e=\frac{\text{Relative velocity of separation along the line of impact}}{\text{Relative velocity of approach along the line of impact}}$

For the collision between ball $A$ and ball $B$ along the line of centers $AB$:
- Before collision, the component of velocity of $A$ along $AB$ is $v_0\cos 30°$, and ball $B$ is at rest. Thus, the relative velocity of approach is:

$u_{\text{approach}}=v_0\cos 30°$

- After collision, ball $A$ is at rest, and ball $B$ moves with velocity $v$ along the line of impact. Thus, the relative velocity of separation is:

$v_{\text{separation}}=v$

Using the definition of $e$:

$e=\frac{v}{v_0\cos 30°}$

Substitute the value of $v$ and $\cos 30°$ into the equation:

$e=\frac{v_0/\sqrt{3}}{v_0\left(\frac{\sqrt{3}}{2}\right)}$

$e=\frac{1}{\sqrt{3}}·\frac{2}{\sqrt{3}}=\frac{2}{3}$

Thus, the coefficient of restitution is indeed $\frac{2}{3}$.