Two bodies with masses `M_(1)` and `M_(2)` are initially at rest and a distance `R` apart. Then they move directly towards one another under the influence of their mutual gravitational attraction. What is the ratio of the distances travelled by `M_(1)` to the distance travelled by `M_(2)`?

Options

A. `(M_(1))/(M_(2))`
B. `(M_(2))/(M_(1))`
C. `1`
D. `1/2`

Explanation

To find the ratio of the distances traveled by two bodies with masses \( M_1 \) and \( M_2 \) under mutual gravitational attraction, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the System**:
- We have two masses, \( M_1 \) and \( M_2 \), initially at rest and a distance \( R \) apart.
- As they move towards each other due to gravitational attraction, we need to analyze their motion.

2. **Defining Distances**:
- Let \( X_1 \) be the distance traveled by mass \( M_1 \) and \( X_2 \) be the distance traveled by mass \( M_2 \).
- The total distance between them remains constant, so we have:
\[ X_1 + X_2 = R \]

3. **Center of Mass Consideration**:
- The center of mass of the system does not move since the only forces acting are internal (gravitational forces between the two masses).
- The position of the center of mass \( R_{cm} \) can be defined as:
\[ R_{cm} = \frac{M_1 \cdot R_1 + M_2 \cdot R_2}{M_1 + M_2} \]
- Here, \( R_1 \) and \( R_2 \) are the distances from a reference point to each mass.

4. **Setting Up the Equation**:
- Let \( R_1 \) be the initial distance of \( M_1 \) from the center of mass and \( R_2 \) be the initial distance of \( M_2 \) from the center of mass.
- We can express the distances in terms of \( X_1 \) and \( X_2 \):
\[ R_1 - X_1 \text{ (for } M_1\text{)} \quad \text{and} \quad R_2 - X_2 \text{ (for } M_2\text{)} \]

5. **Applying the Center of Mass Condition**:
- Since the center of mass does not move, we can write:
\[ M_1(R_1 - X_1) = M_2(R_2 - X_2) \]
- Expanding this gives:
\[ M_1 R_1 - M_1 X_1 = M_2 R_2 - M_2 X_2 \]

6. **Using the Initial Condition**:
- From the initial condition, we know that:
\[ R_1 M_1 = R_2 M_2 \]
- Thus, we can substitute \( R_1 M_1 \) and \( R_2 M_2 \) into our equation, leading to:
\[ -M_1 X_1 + M_2 X_2 = 0 \]
- Rearranging gives:
\[ M_1 X_1 = M_2 X_2 \]

7. **Finding the Ratio**:
- Dividing both sides by \( X_2 \) and \( M_1 \):
\[ \frac{X_1}{X_2} = \frac{M_2}{M_1} \]

### Final Result:
The ratio of the distances traveled by \( M_1 \) to the distance traveled by \( M_2 \) is:
\[ \frac{X_1}{X_2} = \frac{M_2}{M_1} \]