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 \( M_2 \) that are initially at rest and move towards each other due to their mutual gravitational attraction, we can follow these steps:

### Step 1: Understand the System
Two bodies with masses \( M_1 \) and \( M_2 \) are initially at rest and separated by a distance \( R \). They will move towards each other due to the gravitational force acting between them.

### Step 2: Define the Distances
Let \( x_1 \) be the distance traveled by mass \( M_1 \) and \( x_2 \) be the distance traveled by mass \( M_2 \) before they collide.

### Step 3: Center of Mass Consideration
The center of mass of the system will remain at rest since there are no external forces acting on the system. The position of the center of mass \( CM \) can be defined as:
\[ CM = \frac{M_1 x_1 + M_2 x_2}{M_1 + M_2} \]
Since the center of mass does not move, the displacement of the center of mass is zero:
\[ M_1 x_1 + M_2 x_2 = 0 \]

### Step 4: Relate the Distances
From the equation \( M_1 x_1 + M_2 x_2 = 0 \), we can rearrange it to find the relationship between \( x_1 \) and \( x_2 \):
\[ M_1 x_1 = -M_2 x_2 \]
This implies:
\[ M_1 x_1 = M_2 (-x_2) \]
Thus, we can express the ratio of the distances traveled:
\[ \frac{x_1}{x_2} = \frac{M_2}{M_1} \]

### Step 5: Final Ratio
Therefore, the ratio of the distances traveled by \( M_1 \) to the distance traveled by \( M_2 \) is:
\[ \frac{x_1}{x_2} = \frac{M_2}{M_1} \]

### Conclusion
The ratio of the distances traveled by \( M_1 \) to that of \( M_2 \) is \( \frac{M_2}{M_1} \).