Two bodies `A and B` initially at rest are attrached towards each other due to gravitation. Given that `A` is much heavier. Than `B`. Which of the followings correctly describes the relative motion of the centre of mass of the bodies ?

Options

A. it moves towards `A`
B. it moves towards `B`
C. it moves perpendicular to the line joining the particles
D. it remains at rest.

Explanation

To solve the problem, we need to analyze the motion of the center of mass of two bodies, A and B, which are initially at rest and are attracted to each other due to gravitational force.

### Step-by-Step Solution:

1. **Understanding the System**:
- We have two bodies, A and B. Body A is much heavier than body B.
- Both bodies are initially at rest and are only influenced by their mutual gravitational attraction.

2. **Identifying the Forces**:
- The only force acting on the system is the gravitational force between A and B.
- Since A is much heavier than B, the gravitational force will cause B to accelerate towards A.

3. **Applying Newton's Third Law**:
- According to Newton's third law, the force exerted by A on B is equal in magnitude and opposite in direction to the force exerted by B on A.
- This means that while B is pulled towards A, A is also pulled towards B, but the acceleration of A will be much smaller due to its larger mass.

4. **Calculating the Center of Mass**:
- The center of mass (CM) of a two-body system can be calculated using the formula:
\[ x_{CM} = \frac{m_A x_A + m_B x_B}{m_A + m_B} \]
- Here, \(m_A\) and \(m_B\) are the masses of A and B, and \(x_A\) and \(x_B\) are their positions.

5. **Considering the Motion of the Center of Mass**:
- Since the external forces acting on the system are zero (the only forces are internal gravitational forces), the center of mass of the system will not move.
- This is a consequence of the conservation of momentum. The total momentum of the system remains constant, and since the system starts at rest, the center of mass will remain at rest.

6. **Conclusion**:
- Therefore, the correct description of the relative motion of the center of mass of bodies A and B is that it remains at rest.

### Final Answer:
The center of mass of the two bodies A and B remains at rest.

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