Two particles of combined mass M, placed in space with certain separation are released interaction between the particles is only of gravitational nature and there is no external force present. Acceleration of one particle with respect to the other when separation between them is R, has a magnitude :

Options

A. `(GM)/(2R^(2))`
B. `(GM)/(R^(2))`
C. `(2GM)/(R^(2))`
D. not possible to calculate due to lack of information

Explanation

To solve the problem, we need to find the acceleration of one particle with respect to the other when they are separated by a distance \( R \) and only gravitational forces are acting between them.

### Step-by-Step Solution:

1. **Understanding the Forces**:
- Let the two particles have masses \( M_1 \) and \( M_2 \).
- The gravitational force \( F \) between them is given by Newton's law of gravitation:
\[ F = \frac{G M_1 M_2}{R^2} \]
- Here, \( G \) is the gravitational constant, and \( R \) is the separation between the two particles.

2. **Finding Acceleration of Each Particle**:
- The acceleration \( a_1 \) of particle \( M_1 \) due to the gravitational force exerted by \( M_2 \) is given by:
\[ F = M_1 a_1 \implies a_1 = \frac{F}{M_1} = \frac{G M_2}{R^2} \]
- Similarly, the acceleration \( a_2 \) of particle \( M_2 \) due to the gravitational force exerted by \( M_1 \) is:
\[ F = M_2 a_2 \implies a_2 = \frac{F}{M_2} = \frac{G M_1}{R^2} \]

3. **Relative Acceleration**:
- The relative acceleration \( A \) of one particle with respect to the other is the sum of their individual accelerations since they are moving towards each other:
\[ A = a_1 + a_2 = \frac{G M_2}{R^2} + \frac{G M_1}{R^2} \]
- This can be simplified to:
\[ A = \frac{G (M_1 + M_2)}{R^2} \]

4. **Using Combined Mass**:
- Given that the combined mass \( M = M_1 + M_2 \), we can substitute this into our equation:
\[ A = \frac{G M}{R^2} \]

5. **Final Result**:
- Therefore, the magnitude of the acceleration of one particle with respect to the other when the separation between them is \( R \) is:
\[ A = \frac{G M}{R^2} \]