A sphere P of mass m moving with velocity u collides head on with another sphere Q of mass m which is at rest . The ratio of final velocity of Q to initial velocity of P is
( e = coefficient of restitution )

Options

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

Answer

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

### Step-by-Step Solution:

1. **Identify the initial conditions:**
- Sphere P has mass \( m \) and initial velocity \( u \) (i.e., \( u_1 = u \)).
- Sphere Q has mass \( m \) and is at rest (i.e., \( u_2 = 0 \)).

2. **Apply the conservation of momentum:**
The total momentum before the collision must equal the total momentum after the collision. This can be expressed as:
\[ m \cdot u + m \cdot 0 = m \cdot v_1 + m \cdot v_2 \]
Simplifying this gives:
\[ mu = mv_1 + mv_2 \]
Dividing through by \( m \) (assuming \( m \neq 0 \)):
\[ u = v_1 + v_2 \quad \text{(Equation 1)} \]

3. **Use the definition of the coefficient of restitution (e):**
The coefficient of restitution \( e \) is defined as:
\[ e = \frac{\text{relative velocity after collision}}{\text{relative velocity before collision}} \]
For our case:
\[ e = \frac{v_2 - v_1}{u - 0} = \frac{v_2 - v_1}{u} \]
Rearranging gives:
\[ v_2 - v_1 = e \cdot u \quad \text{(Equation 2)} \]

4. **Solve the system of equations:**
We have two equations:
- From Equation 1: \( v_1 + v_2 = u \)
- From Equation 2: \( v_2 - v_1 = e \cdot u \)

To solve for \( v_2 \), we can add these two equations:
\[ (v_1 + v_2) + (v_2 - v_1) = u + e \cdot u \]
This simplifies to:
\[ 2v_2 = u(1 + e) \]
Therefore:
\[ v_2 = \frac{u(1 + e)}{2} \]

5. **Find the ratio of the final velocity of Q to the initial velocity of P:**
The ratio of the final velocity of Q (\( v_2 \)) to the initial velocity of P (\( u \)) is:
\[ \frac{v_2}{u} = \frac{\frac{u(1 + e)}{2}}{u} = \frac{1 + e}{2} \]

### Final Answer:
The ratio of the final velocity of Q to the initial velocity of P is:
\[ \frac{1 + e}{2} \]