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

Answer

To solve the problem, we will follow the principles of conservation of momentum and the definition of the coefficient of restitution.

### Step-by-Step Solution:

1. **Identify Initial Conditions:**
- Mass of both balls = m
- Initial velocity of the first ball (moving) = \(2v_0\)
- Initial velocity of the second ball (at rest) = \(0\)

2. **Apply Conservation of Momentum:**
The total momentum before the collision must equal the total momentum after the collision.
\[ m \cdot 2v_0 + m \cdot 0 = m \cdot v_1 + m \cdot v_2 \]
Simplifying this gives:
\[ 2v_0 = v_1 + v_2 \quad \text{(Equation 1)} \]

3. **Define Coefficient of Restitution (e):**
The coefficient of restitution \(e\) is defined as:
\[ e = \frac{\text{Relative velocity of separation}}{\text{Relative velocity of approach}} \]
Before the collision, the relative velocity of approach is \(2v_0 - 0 = 2v_0\).
After the collision, the relative velocity of separation is \(v_2 - v_1\).
Thus, we can write:
\[ e = \frac{v_2 - v_1}{2v_0} \quad \text{(Equation 2)} \]

4. **Rearranging Equation 2:**
From Equation 2, we can express the relationship between \(v_1\) and \(v_2\):
\[ v_2 - v_1 = 2ev_0 \]
Rearranging gives:
\[ v_2 = v_1 + 2ev_0 \quad \text{(Equation 3)} \]

5. **Substituting Equation 3 into Equation 1:**
Now, substitute \(v_2\) from Equation 3 into Equation 1:
\[ 2v_0 = v_1 + (v_1 + 2ev_0) \]
Simplifying this:
\[ 2v_0 = 2v_1 + 2ev_0 \]
Dividing the entire equation by 2:
\[ v_0 = v_1 + ev_0 \]
Rearranging gives:
\[ v_1 = v_0(1 - e) \quad \text{(Equation 4)} \]

6. **Finding \(v_2\):**
Substitute \(v_1\) from Equation 4 back into Equation 3:
\[ v_2 = v_1 + 2ev_0 = v_0(1 - e) + 2ev_0 \]
Simplifying gives:
\[ v_2 = v_0(1 + e) \quad \text{(Equation 5)} \]

7. **Finding the Ratio of Velocities:**
Now, we can find the ratio of the velocities of the two balls after the collision:
\[ \frac{v_1}{v_2} = \frac{v_0(1 - e)}{v_0(1 + e)} = \frac{1 - e}{1 + e} \]

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