A sphere of mass `m` moving with constant velocity `u`, collides with another stationary sphere of same mass. If `e` is the coefficient of restitution, the ratio of the final velocities of the first and second sphere is

Options

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

Explanation

To solve the problem, we need to analyze the collision between two spheres of equal mass, where one is moving with an initial velocity \( u \) and the other is stationary. We will use the principles of conservation of momentum and the coefficient of restitution.

### Step-by-Step Solution:

1. **Identify the Variables:**
- Mass of both spheres: \( m \)
- Initial velocity of the first sphere: \( u \)
- Initial velocity of the second sphere: \( 0 \) (stationary)
- Final velocity of the first sphere after collision: \( v_1 \)
- Final velocity of the second sphere after collision: \( v_2 \)
- Coefficient of restitution: \( e \)

2. **Apply Conservation of Momentum:**
Since there are no external forces acting on the system, the total momentum before the collision must equal the total momentum after the collision.
\[ mu + 0 = mv_1 + mv_2 \]
Simplifying this, we get:
\[ u = v_1 + v_2 \quad \text{(Equation 1)} \]

3. **Apply the Coefficient of Restitution:**
The coefficient of restitution \( e \) is defined as the ratio of the relative velocity of separation to the relative velocity of approach.
\[ e = \frac{\text{Relative velocity of separation}}{\text{Relative velocity of approach}} \]
The relative velocity of approach is \( u \) (the first sphere approaches the second sphere at speed \( u \)). The relative velocity of separation is \( v_2 - v_1 \) (the second sphere moves away from the first sphere after the collision). Thus, we can write:
\[ e = \frac{v_2 - v_1}{u} \quad \text{(Equation 2)} \]

4. **Rearranging Equation 2:**
Rearranging Equation 2 gives us:
\[ v_2 - v_1 = eu \quad \Rightarrow \quad v_2 = v_1 + eu \quad \text{(Equation 3)} \]

5. **Substituting Equation 3 into Equation 1:**
Now we can substitute Equation 3 into Equation 1:
\[ u = v_1 + (v_1 + eu) \]
Simplifying this, we have:
\[ u = 2v_1 + eu \]
Rearranging gives:
\[ u - eu = 2v_1 \quad \Rightarrow \quad v_1 = \frac{u(1 - e)}{2} \quad \text{(Equation 4)} \]

6. **Finding \( v_2 \):**
Now we can find \( v_2 \) using Equation 3:
\[ v_2 = v_1 + eu = \frac{u(1 - e)}{2} + eu \]
Simplifying this, we get:
\[ v_2 = \frac{u(1 - e)}{2} + \frac{2eu}{2} = \frac{u(1 - e + 2e)}{2} = \frac{u(1 + e)}{2} \quad \text{(Equation 5)} \]

7. **Finding the Ratio \( \frac{v_1}{v_2} \):**
Now we can find the ratio of the final velocities:
\[ \frac{v_1}{v_2} = \frac{\frac{u(1 - e)}{2}}{\frac{u(1 + e)}{2}} = \frac{1 - e}{1 + e} \]

### Final Answer:
The ratio of the final velocities of the first sphere to the second sphere is:
\[ \frac{v_1}{v_2} = \frac{1 - e}{1 + e} \]