Ball 1 collides directly with another identical ball 2 at rest. Velocity of second ball becomes two times that of 1 after collison. Find the coefficient of restitution between the two balls?

Explanation

To solve the problem, we need to find the coefficient of restitution (e) between the two balls after a collision. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the scenario
- We have two identical balls: Ball 1 (mass m) is moving with an initial velocity (u) and collides with Ball 2 (mass m) which is initially at rest (velocity = 0).
- After the collision, the velocity of Ball 2 becomes twice that of Ball 1.

### Step 2: Set up the equations
- Let the initial velocity of Ball 1 be \( u \).
- Let the final velocity of Ball 1 after the collision be \( v_1 \).
- The final velocity of Ball 2 after the collision is given as \( v_2 = 2v_1 \).

### Step 3: Apply the conservation of momentum
- The total momentum before the collision is equal to the total momentum after the collision.
- Before the collision:
\[ \text{Total momentum} = mu + 0 = mu \]
- After the collision:
\[ \text{Total momentum} = mv_1 + mv_2 = mv_1 + m(2v_1) = mv_1 + 2mv_1 = 3mv_1 \]
- Setting the two equal gives:
\[ mu = 3mv_1 \]
- Dividing both sides by \( m \) (assuming \( m \neq 0 \)):
\[ u = 3v_1 \quad \text{(1)} \]

### Step 4: Calculate 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:
\[ v_2 - v_1 = 2v_1 - v_1 = v_1 \]
- The velocity of approach before the collision is:
\[ u - 0 = u \]
- Substituting these into the formula for e gives:
\[ e = \frac{v_1}{u} \]

### Step 5: Substitute from equation (1)
- From equation (1), we have \( u = 3v_1 \).
- Therefore:
\[ e = \frac{v_1}{3v_1} = \frac{1}{3} \]

### Final Answer
The coefficient of restitution \( e \) between the two balls is \( \frac{1}{3} \).