10 Two particles of masses `m_(1)` and `m_(2)` initially at rest start moving towards each other under their mutual force of attraction. The speed of the centre of mass at any time t, when they are at a distance r apart, is

Options

A. zero
B. `(G(m_(1)m_(2)))/(r^(2)).(1)/(m_(1))t`
C. `(G.(m_(1)m_(2)))/(r^(2)).(1)/(m_(2))t`
D. `(G(m_(1)m_(2))/(t^(2)).(1)/(m_(1)+m_(2)))t`

Answer

To solve the problem, we need to determine the speed of the center of mass of two particles of masses \( m_1 \) and \( m_2 \) that are initially at rest and moving towards each other under their mutual force of attraction.

### Step-by-Step Solution:

1. **Understanding the System**:
- We have two particles with masses \( m_1 \) and \( m_2 \).
- Initially, both particles are at rest, which means their initial velocities \( u_1 = 0 \) and \( u_2 = 0 \).

2. **Defining the Center of Mass**:
- The center of mass \( R \) of the system can be defined as:
\[ R = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2} \]
- Where \( x_1 \) and \( x_2 \) are the positions of \( m_1 \) and \( m_2 \) respectively.

3. **Applying Conservation of Momentum**:
- According to the conservation of momentum, the total momentum before any interaction must equal the total momentum after:
\[ m_1 u_1 + m_2 u_2 = (m_1 + m_2) v \]
- Since both particles are initially at rest, we have:
\[ 0 + 0 = (m_1 + m_2) v \]
- This simplifies to:
\[ 0 = (m_1 + m_2) v \]
- Therefore, the velocity of the center of mass \( v \) is:
\[ v = 0 \]

4. **Considering Forces Acting on the System**:
- As the particles move towards each other due to their mutual gravitational attraction, the forces acting on each particle are equal in magnitude and opposite in direction (Newton's Third Law).
- The force on \( m_1 \) is equal to the force on \( m_2 \), thus the net external force acting on the center of mass is zero.

5. **Conclusion**:
- Since there is no net external force acting on the center of mass, its velocity remains constant. Since we found that the initial velocity was zero, the speed of the center of mass remains zero throughout the motion.

Thus, the speed of the center of mass at any time \( t \), when the two particles are at a distance \( r \) apart, is:
\[ \text{Speed of center of mass} = 0 \]

### Answer:
The correct option is **A: 0**.