Sphere A of mass `m` moving with a constant velocity `u` hits another stationary sphere B of the same mass. If `e` is the co-efficient of restitution, then ratio of velocities of the two spheres `v_(A):v_(B)` after collision will be :

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 will analyze the collision of two spheres using the principles of conservation of momentum and the coefficient of restitution.

### Step-by-Step Solution:

1. **Identify the Initial Conditions:**
- Sphere A has mass \( m \) and initial velocity \( u \).
- Sphere B has mass \( m \) and is stationary, so its initial velocity is \( 0 \).

2. **Calculate Initial Momentum:**
- The initial momentum \( P_i \) of the system is given by:
\[ P_i = m \cdot u + m \cdot 0 = m \cdot u \]

3. **Define Final Velocities:**
- Let \( v_A \) be the final velocity of sphere A after the collision.
- Let \( v_B \) be the final velocity of sphere B after the collision.

4. **Calculate Final Momentum:**
- The final momentum \( P_f \) of the system is:
\[ P_f = m \cdot v_A + m \cdot v_B = m(v_A + v_B) \]

5. **Apply Conservation of Momentum:**
- Since no external forces are acting on the system, we can equate the initial and final momentum:
\[ m \cdot u = m(v_A + v_B) \]
- Dividing through by \( m \) (assuming \( m \neq 0 \)):
\[ u = v_A + v_B \quad \text{(1)} \]

6. **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 separation is \( v_B - v_A \) and the relative velocity of approach is \( u \):
\[ e = \frac{v_B - v_A}{u} \quad \text{(2)} \]

7. **Rearranging Equation (2):**
- From equation (2), we can express \( v_B \):
\[ v_B - v_A = e \cdot u \]
\[ v_B = e \cdot u + v_A \quad \text{(3)} \]

8. **Substituting into Equation (1):**
- Substitute equation (3) into equation (1):
\[ u = v_A + (e \cdot u + v_A) \]
\[ u = 2v_A + e \cdot u \]
- Rearranging gives:
\[ u - e \cdot u = 2v_A \]
\[ u(1 - e) = 2v_A \]
\[ v_A = \frac{u(1 - e)}{2} \quad \text{(4)} \]

9. **Finding \( v_B \):**
- Substitute \( v_A \) from equation (4) back into equation (3):
\[ v_B = e \cdot u + \frac{u(1 - e)}{2} \]
\[ v_B = e \cdot u + \frac{u}{2} - \frac{e \cdot u}{2} \]
\[ v_B = \frac{u}{2}(1 + e) \quad \text{(5)} \]

10. **Finding the Ratio of Velocities:**
- Now we can find the ratio \( \frac{v_A}{v_B} \):
\[ \frac{v_A}{v_B} = \frac{\frac{u(1 - e)}{2}}{\frac{u(1 + e)}{2}} = \frac{1 - e}{1 + e} \]

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