Two billiard balls of same size and mass are in contact on a billiard table. A third ball of same mass and size strikes them symmetrically and remains at rest after the impact. Find the coefficient of restitution between the balls?

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the scenario
We have two billiard balls of the same mass \( m \) in contact on a table, and a third ball of the same mass \( m \) strikes them symmetrically. After the impact, the third ball remains at rest. We need to find the coefficient of restitution \( e \) between the balls.

### Step 2: Define the initial conditions
Let:
- The initial velocity of the third ball (the one that strikes) be \( u \).
- The two balls in contact are initially at rest.

### Step 3: Analyze the collision
Since the third ball strikes the two balls symmetrically, we can conclude that the angle between the lines connecting the centers of the two balls is \( 60^\circ \) (as they form an equilateral triangle).

### Step 4: Apply conservation of momentum
Before the collision, the total momentum in the horizontal direction is:
\[ p_{\text{initial}} = mu \]
After the collision, let \( V \) be the velocity of each of the two balls after the collision. The momentum of the two balls after the collision is:
\[ p_{\text{final}} = 2mV \cos(30^\circ) \]
Using the conservation of momentum:
\[ mu = 2mV \cos(30^\circ) \]
Dividing through by \( m \):
\[ u = 2V \cdot \frac{\sqrt{3}}{2} \]
This simplifies to:
\[ u = V\sqrt{3} \]
Thus, we find:
\[ V = \frac{u}{\sqrt{3}} \]

### Step 5: Use the definition of the coefficient of restitution
The coefficient of restitution \( e \) is defined as:
\[ e = \frac{\text{Relative velocity after collision}}{\text{Relative velocity before collision}} \]
The relative velocity before the collision (in the direction of the line of impact) is:
\[ u_{\text{before}} = u - 0 = u \]
The relative velocity after the collision for the two balls is:
\[ u_{\text{after}} = V - 0 = V \]
Thus, we can write:
\[ e = \frac{V}{u} \]

### Step 6: Substitute the value of \( V \)
Substituting \( V = \frac{u}{\sqrt{3}} \) into the equation for \( e \):
\[ e = \frac{\frac{u}{\sqrt{3}}}{u} = \frac{1}{\sqrt{3}} \]

### Step 7: Final answer
The coefficient of restitution between the balls is:
\[ e = \frac{1}{\sqrt{3}} \approx 0.577 \]