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 of finding the ratio of velocities of two balls after a head-on collision, we can follow these steps:

### Step 1: Understand the Initial Conditions
We have two identical balls, each with mass \( m \). One ball is moving with an initial speed of \( 2v_0 \), and the other ball is at rest.

### Step 2: Apply the Law of Conservation of Momentum
Before the collision, the total momentum of the system is:
\[ \text{Initial Momentum} = m(2v_0) + m(0) = 2mv_0 \]
After the collision, let the velocities of the two balls be \( v_1 \) (for the ball that was initially moving) and \( v_2 \) (for the ball that was initially at rest). The total momentum after the collision can be expressed as:
\[ \text{Final Momentum} = mv_1 + mv_2 \]
According to the conservation of momentum:
\[ 2mv_0 = mv_1 + mv_2 \]
Dividing through by \( m \):
\[ 2v_0 = v_1 + v_2 \quad \text{(Equation 1)} \]

### Step 3: Apply the Coefficient of Restitution
The coefficient of restitution \( e \) is defined as:
\[ e = \frac{v_2 - v_1}{v_1 + v_2} \]
Rearranging this gives:
\[ v_2 - v_1 = e(v_1 + v_2) \]
Substituting \( v_1 + v_2 \) from Equation 1 into this equation:
\[ v_2 - v_1 = e(2v_0) \]
This can be rearranged to:
\[ v_2 = v_1 + e(2v_0) \quad \text{(Equation 2)} \]

### Step 4: Substitute Equation 2 into Equation 1
Now, substitute Equation 2 into Equation 1:
\[ 2v_0 = v_1 + (v_1 + e(2v_0)) \]
This simplifies to:
\[ 2v_0 = 2v_1 + e(2v_0) \]
Rearranging gives:
\[ 2v_0 - e(2v_0) = 2v_1 \]
Factoring out \( 2v_0 \):
\[ 2v_0(1 - e) = 2v_1 \]
Dividing both sides by 2:
\[ v_0(1 - e) = v_1 \quad \text{(Equation 3)} \]

### Step 5: Find \( v_2 \) Using Equation 3
Now, substitute Equation 3 back into Equation 2 to find \( v_2 \):
\[ v_2 = v_1 + e(2v_0) = v_0(1 - e) + e(2v_0) \]
This simplifies to:
\[ v_2 = v_0(1 - e + 2e) = v_0(1 + e) \quad \text{(Equation 4)} \]

### Step 6: Calculate the Ratio of Velocities
Now we have expressions for both \( v_1 \) and \( v_2 \):
- \( v_1 = v_0(1 - e) \)
- \( v_2 = v_0(1 + e) \)

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

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