Gravitational force with which a body attracts the other is always equal to the force with which the other attracts the first. Assuming no other forces acting on the bodies, choose the correct statement.

Options

A. Both the bodies always have equal acceleration
B. Both the bodies may have equal acceleration
C. Both the bodies may not have equal acceleration
D. The acceleration of the centre of mass of the two bodies is zero

Explanation

To solve the question regarding gravitational force and the accelerations of two bodies, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Gravitational Force**: According to Newton's law of gravitation, the gravitational force \( F \) between two masses \( m_1 \) and \( m_2 \) separated by a distance \( r \) is given by:
\[ F = \frac{G m_1 m_2}{r^2} \]
where \( G \) is the gravitational constant.

**Hint**: Remember that gravitational force is mutual; each mass attracts the other with equal magnitude.

2. **Applying Newton's Second Law**: The acceleration \( a \) of each mass due to the gravitational force can be calculated using Newton's second law, which states that force equals mass times acceleration:
- For mass \( m_1 \):
\[ F_{12} = m_1 a_1 \implies a_1 = \frac{F_{12}}{m_1} = \frac{G m_1 m_2}{m_1 r^2} = \frac{G m_2}{r^2} \]
- For mass \( m_2 \):
\[ F_{21} = m_2 a_2 \implies a_2 = \frac{F_{21}}{m_2} = \frac{G m_1 m_2}{m_2 r^2} = \frac{G m_1}{r^2} \]

**Hint**: Use Newton's second law to relate force and acceleration for each mass.

3. **Comparing Accelerations**: From the above equations, we can see that:
- \( a_1 = \frac{G m_2}{r^2} \)
- \( a_2 = \frac{G m_1}{r^2} \)

These accelerations will only be equal if \( m_1 = m_2 \). If the masses are different, the accelerations will not be equal.

**Hint**: Equal accelerations occur only when the masses are equal.

4. **Acceleration of the Center of Mass**: The acceleration of the center of mass \( a_{cm} \) of a system of two bodies is given by:
\[ a_{cm} = \frac{m_1 a_1 + m_2 a_2}{m_1 + m_2} \]
Since the gravitational force between the two bodies is an internal force, it does not affect the acceleration of the center of mass. Therefore, the acceleration of the center of mass is zero if no external forces act on the system.

**Hint**: Internal forces do not change the motion of the center of mass.

5. **Conclusion**: Based on the analysis:
- The gravitational force between two bodies is mutual and equal in magnitude.
- The bodies may have equal accelerations only if their masses are equal.
- The accelerations of the two bodies may not be equal if their masses differ.
- The acceleration of the center of mass of the system is zero due to the internal nature of gravitational forces.

### Final Answer:
The correct statements are:
- Both bodies may have equal acceleration (if \( m_1 = m_2 \)).
- Both bodies may not have equal acceleration (if \( m_1 \neq m_2 \)).
- The acceleration of the center of mass of the two bodies is zero.

Thus, options B, C, and D are correct.