A body of mass `m` is dropped and another body of mass `M` is projected vertically up with speed `u` simultaneously from the top of a tower of height `H`. If the body reaches the highest piont before the dropped body reaches the ground, then maximum height raised by the centre of mass of the system from ground is

Options

A. `H+(u^(2))/(2g)`
B. `(u^(2))/(2g)`
C. `H+1/(2g)((Mu)/(m+M))^(2)`
D. `H+1/(2g)((m u)/(m+M))^(2)`

Explanation

To solve the problem, we need to determine the maximum height raised by the center of mass of the system consisting of two bodies: one of mass `m` that is dropped and another of mass `M` that is projected upwards with speed `u` from the top of a tower of height `H`.

### Step-by-Step Solution:

1. **Identify Initial Conditions:**
- Mass of the body dropped: `m`
- Mass of the body projected upwards: `M`
- Initial height of the tower: `H`
- Initial velocity of the dropped body: `0` (since it is dropped)
- Initial velocity of the projected body: `u` (upwards)

2. **Determine the Center of Mass (CM) Position:**
The initial position of the center of mass (CM) of the system can be calculated using the formula:
\[ x_{CM} = \frac{m \cdot 0 + M \cdot H}{m + M} = \frac{M \cdot H}{m + M} \]
Here, the dropped body is at the ground level (0) and the projected body is at height `H`.

3. **Calculate the Velocity of the Center of Mass:**
The initial velocity of the center of mass (CM) can be calculated as:
\[ v_{CM} = \frac{m \cdot 0 + M \cdot u}{m + M} = \frac{M \cdot u}{m + M} \]

4. **Determine the Time Until the Projected Body Reaches Maximum Height:**
The time taken for the body of mass `M` to reach its maximum height can be calculated using the formula:
\[ t = \frac{u}{g} \]
where `g` is the acceleration due to gravity.

5. **Calculate the Maximum Height Reached by the Projected Body:**
The maximum height reached by the body of mass `M` can be calculated using the formula:
\[ H_{max} = H + \frac{u^2}{2g} \]

6. **Calculate the Maximum Height of the Center of Mass:**
The maximum height of the center of mass when the projected body reaches its maximum height can be calculated as:
\[ H_{CM} = H + \frac{M \cdot u^2}{2g(m + M)} \]

7. **Final Expression for Maximum Height of the Center of Mass:**
Thus, the maximum height raised by the center of mass from the ground is:
\[ H_{CM} = H + \frac{M \cdot u^2}{2g(m + M)} \]

### Final Answer:
The maximum height raised by the center of mass of the system from the ground is:
\[ H + \frac{M \cdot u^2}{2g(m + M)} \]