A uniform rod of length `L` lies on a smooth horizontal table. The rod has a mass `M`. A particle of mass `m` moving with speed `v` strikes the rod perpendicularly at one of the ends of the rod sticks to it after collision.
Find the velocity of the particle with respect to `C` before the collision

Options

A. `(Mv)/(M+m)`
B. `(mv)/(M+m)`
C. `(2mv)/(M+m)`
D. `(2Mv)/(M+m)`

Explanation

To solve the problem, we need to find the velocity of the particle of mass `m` with respect to the center of mass (C) of the system (particle + rod) before the collision. Here are the steps to arrive at the solution:

### Step-by-Step Solution:

1. **Identify the System**: We have a uniform rod of mass `M` and a particle of mass `m` moving with speed `v`. The rod is initially at rest on a smooth horizontal table.

2. **Determine the Center of Mass (C)**: The center of mass of the system can be calculated using the formula:
\[ V_{cm} = \frac{m_1 v_1 + m_2 v_2}{m_1 + m_2} \]
Here, `m_1` is the mass of the particle (`m`), `v_1` is its velocity (`v`), `m_2` is the mass of the rod (`M`), and `v_2` is its velocity (which is `0` since the rod is at rest).

3. **Substituting Values**: Plugging in the values, we get:
\[ V_{cm} = \frac{m \cdot v + M \cdot 0}{m + M} = \frac{mv}{m + M} \]

4. **Finding the Velocity of the Particle with Respect to C**: The velocity of the particle `m` with respect to the center of mass `C` is given by:
\[ V_{m, C} = V_m - V_{cm} \]
where `V_m` is the velocity of the particle before the collision (`v`).

5. **Substituting for `V_{cm}`**: Now substituting for `V_{cm}`:
\[ V_{m, C} = v - \frac{mv}{m + M} \]

6. **Finding a Common Denominator**: To simplify this expression, we can write:
\[ V_{m, C} = \frac{v(m + M) - mv}{m + M} \]

7. **Simplifying the Expression**: This simplifies to:
\[ V_{m, C} = \frac{vM}{m + M} \]

### Final Answer:
Thus, the velocity of the particle of mass `m` with respect to the center of mass `C` before the collision is:
\[ V_{m, C} = \frac{vM}{m + M} \]