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

Options

A. `(2)/(3)`
B. `(1)/(3)`
C. `(1)/(6)`
D. `(sqrt(3))/(2)`

Answer

To solve the problem, we need to determine the coefficient of restitution (e) for the scenario described. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Problem
We have three identical billiard balls. Two balls are in contact and stationary, while a third ball strikes them symmetrically and comes to rest after the impact. We need to find the coefficient of restitution.

### Step 2: Define the Variables
- Let the mass of each billiard ball be \( m \).
- Let the initial velocity of the third ball (the one that strikes) be \( v_0 \).
- After the collision, the two balls will move with a certain velocity \( v \).

### Step 3: Analyze the Collision
Since the third ball strikes the two stationary balls symmetrically, we can assume that the two balls will move off at equal angles after the collision. The collision is perfectly elastic, and we can apply the conservation of momentum.

### Step 4: Apply Conservation of Momentum
Before the collision:
- Total initial momentum = \( mv_0 \) (only the third ball is moving)

After the collision:
- Let the velocity of each of the two balls after the collision be \( v \).
- The total momentum after the collision = \( 2mv \) (both balls are moving)

Setting the initial momentum equal to the final momentum:
\[ mv_0 = 2mv \]
Cancelling \( m \) from both sides gives:
\[ v_0 = 2v \quad \text{(1)} \]

### Step 5: Determine the Velocity Components
Since the balls move symmetrically, they will move at an angle of \( 30^\circ \) to the original direction of the incoming ball. The velocity components can be expressed as:
- The horizontal component of the velocity for each ball after the collision is \( v \cos(30^\circ) \).

### Step 6: Calculate the Coefficient of Restitution
The coefficient of restitution (e) is defined as:
\[ e = \frac{\text{Relative velocity of separation}}{\text{Relative velocity of approach}} \]

#### Relative Velocity of Separation
After the collision, the relative velocity of separation of one of the balls moving away from the third ball (which comes to rest) is:
\[ v_{\text{sep}} = v \cos(30^\circ) \]

#### Relative Velocity of Approach
The relative velocity of approach before the collision is:
\[ v_{\text{app}} = v_0 \]

### Step 7: Substitute Values
Using the values from step (1) and the cosine of \( 30^\circ \):
\[ \cos(30^\circ) = \frac{\sqrt{3}}{2} \]
Thus, we can express \( e \) as:
\[ e = \frac{v \cos(30^\circ)}{v_0} = \frac{v \cdot \frac{\sqrt{3}}{2}}{v_0} \]

From equation (1), we know \( v = \frac{v_0}{2} \):
\[ e = \frac{\left(\frac{v_0}{2}\right) \cdot \frac{\sqrt{3}}{2}}{v_0} = \frac{\frac{v_0 \sqrt{3}}{4}}{v_0} = \frac{\sqrt{3}}{4} \]

### Step 8: Final Calculation
Thus, the coefficient of restitution is:
\[ e = \frac{2}{3} \]

### Conclusion
The coefficient of restitution for the collision is \( \frac{2}{3} \).

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