In a free space a rifle on mass `M` shoots a bullet of mass `m` at a stationary block of mass `M` distance `D` away from it . When the bullet has moved through a distance `d` towards the block the centre of mass of the bullet `-` block system is at a distance of `:`

Options

A. `((D-d)m)/(M +m)` from the bullet
B. `(md + MD)/(M + m)` from the block
C. `(2md + MD)/(M + m)`from the block
D. `((D-d)M)/(M + m)` from the bullet

Explanation

To solve the problem, we need to find the position of the center of mass of the bullet-block system when the bullet has moved a distance \( d \) towards the block.

### Step-by-Step Solution:

1. **Define the System**:
- Let the mass of the bullet be \( m \).
- Let the mass of the block be \( M \).
- The initial distance between the rifle (where the bullet is shot from) and the block is \( D \).
- When the bullet has moved a distance \( d \) towards the block, the new distance between the bullet and the block becomes \( D - d \).

2. **Position of the Bullet and Block**:
- Initially, we can consider the position of the bullet as \( 0 \) (the rifle's position) and the position of the block as \( D \).
- After the bullet has moved a distance \( d \), the position of the bullet becomes \( d \) and the position of the block remains \( D \).

3. **Calculate the Center of Mass**:
- The formula for the center of mass \( R_{cm} \) of a system of particles is given by:
\[ R_{cm} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2} \]
- Here, \( m_1 = m \) (mass of the bullet), \( x_1 = d \) (position of the bullet), \( m_2 = M \) (mass of the block), and \( x_2 = D \) (position of the block).
- Substituting these values into the formula gives:
\[ R_{cm} = \frac{m \cdot d + M \cdot D}{m + M} \]

4. **Interpret the Result**:
- The center of mass \( R_{cm} \) gives us the position of the center of mass of the bullet-block system after the bullet has moved a distance \( d \) towards the block.

5. **Final Expression**:
- Thus, the center of mass of the bullet-block system when the bullet has moved a distance \( d \) towards the block is:
\[ R_{cm} = \frac{m \cdot d + M \cdot D}{m + M} \]