Ball 1 collides with another identical ball at rest. For what value of coefficient of restitution e, the velocity of second ball becomes two times that of 1 after collision? _____.

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the scenario
We have two identical balls. Ball 1 is moving with an initial velocity \( v \), and Ball 2 is at rest. After the collision, we need to find the coefficient of restitution \( e \) such that the velocity of Ball 2 becomes two times that of Ball 1.

### Step 2: Define the velocities after the collision
Let:
- The velocity of Ball 1 after the collision be \( u \).
- The velocity of Ball 2 after the collision be \( 2u \).

### Step 3: Apply the conservation of momentum
According to the conservation of momentum:
\[ \text{Initial momentum} = \text{Final momentum} \]
Before the collision, the total momentum is:
\[ mv + 0 = mv \]
After the collision, the total momentum is:
\[ mu + m(2u) = mu + 2mu = 3mu \]
Setting the initial momentum equal to the final momentum:
\[ mv = 3mu \]
Dividing both sides by \( m \) (since \( m \neq 0 \)):
\[ v = 3u \]
From this, we can express \( u \) in terms of \( v \):
\[ u = \frac{v}{3} \]

### Step 4: Use the definition of the coefficient of restitution
The coefficient of restitution \( e \) is defined as:
\[ e = \frac{\text{Velocity of separation}}{\text{Velocity of approach}} \]
The velocity of separation after the collision is:
\[ \text{Velocity of separation} = 2u - u = u \]
The velocity of approach before the collision is:
\[ \text{Velocity of approach} = v - 0 = v \]
Thus, we can write:
\[ e = \frac{u}{v} \]

### Step 5: Substitute \( u \) in terms of \( v \)
From our earlier result, we have \( u = \frac{v}{3} \). Substituting this into the equation for \( e \):
\[ e = \frac{\frac{v}{3}}{v} = \frac{1}{3} \]

### Conclusion
The value of the coefficient of restitution \( e \) is:
\[ \boxed{\frac{1}{3}} \]