A ball of mass m moving with a speed `2v_0` collides head-on with an identical ball at rest. If e is the coefficient of restitution, then what will be the ratio of velocity of two balls after collision?

Options

A. `(1-e)/(1+e)`
B. `(1+e)/(1-e)`
C. `(e-1)/(e+1)`
D. `(e+1)/(e-1)`

Explanation

To solve the problem, we need to apply the principles of conservation of momentum and the definition of the coefficient of restitution. Let's break it down step by step.

### Step 1: Understand the scenario
We have two identical balls of mass \( m \). One ball is moving with a speed of \( 2v_0 \) and the other ball is at rest. After the collision, we need to find the ratio of their velocities.

### Step 2: Define the variables
Let:
- \( u_1 = 2v_0 \) (initial velocity of the moving ball)
- \( u_2 = 0 \) (initial velocity of the ball at rest)
- \( v_1 \) = final velocity of the first ball after collision
- \( v_2 \) = final velocity of the second ball after collision

### Step 3: Apply conservation of momentum
According to the law of conservation of momentum:
\[ m u_1 + m u_2 = m v_1 + m v_2 \]
Substituting the values:
\[ m(2v_0) + m(0) = m v_1 + m v_2 \]
This simplifies to:
\[ 2v_0 = v_1 + v_2 \quad \text{(1)} \]

### Step 4: Apply the coefficient of restitution
The coefficient of restitution \( e \) is defined as:
\[ e = \frac{\text{relative velocity after collision}}{\text{relative velocity before collision}} \]
Before the collision, the relative velocity of separation is \( v_2 - v_1 \) and the relative velocity of approach is \( u_1 - u_2 = 2v_0 - 0 = 2v_0 \). Therefore:
\[ e = \frac{v_2 - v_1}{2v_0} \quad \text{(2)} \]
Rearranging equation (2):
\[ v_2 - v_1 = 2ev_0 \quad \text{(3)} \]

### Step 5: Solve the equations
Now we have two equations (1) and (3):
1. \( v_1 + v_2 = 2v_0 \)
2. \( v_2 - v_1 = 2ev_0 \)

From equation (1), we can express \( v_2 \) in terms of \( v_1 \):
\[ v_2 = 2v_0 - v_1 \quad \text{(4)} \]

Substituting equation (4) into equation (3):
\[ (2v_0 - v_1) - v_1 = 2ev_0 \]
This simplifies to:
\[ 2v_0 - 2v_1 = 2ev_0 \]
Dividing through by 2:
\[ v_0 - v_1 = ev_0 \]
Rearranging gives:
\[ v_1 = v_0(1 - e) \quad \text{(5)} \]

Now substitute equation (5) back into equation (4) to find \( v_2 \):
\[ v_2 = 2v_0 - v_0(1 - e) = 2v_0 - v_0 + ev_0 = v_0(1 + e) \quad \text{(6)} \]

### Step 6: Find the ratio of the velocities
Now we have:
- \( v_1 = v_0(1 - e) \)
- \( v_2 = v_0(1 + e) \)

The ratio of the velocities after the collision is:
\[ \frac{v_1}{v_2} = \frac{v_0(1 - e)}{v_0(1 + e)} = \frac{1 - e}{1 + e} \]

### Final Answer
The ratio of the velocities of the two balls after the collision is:
\[ \frac{v_1}{v_2} = \frac{1 - e}{1 + e} \]