The first ball of mass `m` moving with the velocity `upsilon` collides head on with the second ball of mass `m` at rest. If the coefficient of restitution is `e`, then the ratio of the velocities of the first and the second ball after the collision is

Options

A. `(1 -e)/(1 + e)`
B. `(1 + e)/(1 - e)`
C. `(1 + e)/2`
D. `(1 - e)/2`

Explanation

To solve the problem, we will analyze the collision between two balls using the principles of conservation of momentum and the coefficient of restitution.

### Step-by-Step Solution:

1. **Identify the Initial Conditions:**
- Let the mass of both balls be \( m \).
- The first ball (Ball A) is moving with an initial velocity \( v \).
- The second ball (Ball B) is at rest, so its initial velocity \( u = 0 \).

2. **Define the Final Velocities:**
- After the collision, let the velocity of Ball A be \( v_1 \) and the velocity of Ball B be \( v_2 \).

3. **Apply the Conservation of Momentum:**
- The total momentum before the collision must equal the total momentum after the collision.
\[ mv = mv_1 + mv_2 \]
- Dividing through by \( m \) (since \( m \neq 0 \)):
\[ v = v_1 + v_2 \quad \text{(Equation 1)} \]

4. **Use the Coefficient of Restitution:**
- The coefficient of restitution \( e \) is defined as:
\[ e = \frac{\text{Relative velocity of separation}}{\text{Relative velocity of approach}} \]
- The relative velocity of approach is \( v \) (since Ball A approaches Ball B).
- The relative velocity of separation is \( v_2 - v_1 \) (Ball B moves away faster than Ball A).
\[ e = \frac{v_2 - v_1}{v} \quad \text{(Equation 2)} \]

5. **Rearranging Equation 2:**
- From Equation 2, we can express it as:
\[ ev = v_2 - v_1 \]
- Rearranging gives:
\[ v_2 = ev + v_1 \quad \text{(Equation 3)} \]

6. **Substituting Equation 3 into Equation 1:**
- Substitute \( v_2 \) from Equation 3 into Equation 1:
\[ v = v_1 + (ev + v_1) \]
- Simplifying this gives:
\[ v = v_1 + ev + v_1 \]
\[ v = 2v_1 + ev \]
- Rearranging gives:
\[ v(1 - e) = 2v_1 \]
\[ v_1 = \frac{v(1 - e)}{2} \quad \text{(Equation 4)} \]

7. **Finding \( v_2 \):**
- Now substitute \( v_1 \) back into Equation 3 to find \( v_2 \):
\[ v_2 = ev + \frac{v(1 - e)}{2} \]
- Simplifying gives:
\[ v_2 = ev + \frac{v - ev}{2} = ev + \frac{v}{2} - \frac{ev}{2} \]
\[ v_2 = \frac{v}{2} + \frac{ev}{2} = \frac{v(1 + e)}{2} \quad \text{(Equation 5)} \]

8. **Finding the Ratio \( \frac{v_1}{v_2} \):**
- Now, we can find the ratio of the velocities:
\[ \frac{v_1}{v_2} = \frac{\frac{v(1 - e)}{2}}{\frac{v(1 + e)}{2}} \]
- The \( \frac{v}{2} \) cancels out:
\[ \frac{v_1}{v_2} = \frac{1 - e}{1 + e} \]

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