Two particles of mass `m_(1)` and `m_(2)` are approaching towards each other under their mutual gravitational field. If the speed of the particle is v, the speed of the center of mass of the system is equal to :

Options

A. `(m_(1))/(m_(2))v`
B. Zero
C. `((m_(1)-m_(2))v)/(m_(1)+m_(2))`
D. `(m_(2))/(m_(1))v`

Answer

To find the speed of the center of mass of a system of two particles moving towards each other, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Masses and Velocities**:
- Let the mass of the first particle be \( m_1 \).
- Let the mass of the second particle be \( m_2 \).
- Both particles are moving towards each other with a speed \( v \).

2. **Assign Directions**:
- Assume that particle \( m_1 \) is moving in the positive direction with speed \( v \).
- Therefore, particle \( m_2 \) will be moving in the negative direction with speed \( -v \).

3. **Use the Formula for Center of Mass Velocity**:
- The velocity of the center of mass \( V_{cm} \) of a system of particles is given by the formula:
\[ V_{cm} = \frac{\sum (m_i v_i)}{\sum m_i} \]
- For our two particles, this becomes:
\[ V_{cm} = \frac{m_1 v + m_2 (-v)}{m_1 + m_2} \]

4. **Substitute the Values**:
- Substituting the velocities into the equation:
\[ V_{cm} = \frac{m_1 v - m_2 v}{m_1 + m_2} \]

5. **Factor Out the Common Term**:
- Factor \( v \) out of the numerator:
\[ V_{cm} = \frac{(m_1 - m_2)v}{m_1 + m_2} \]

6. **Final Expression**:
- Thus, the speed of the center of mass of the system is:
\[ V_{cm} = \frac{(m_1 - m_2)v}{m_1 + m_2} \]

### Conclusion:
The speed of the center of mass of the two particles is given by:
\[ V_{cm} = \frac{(m_1 - m_2)v}{m_1 + m_2} \]