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Step-by-Step Solution
Step 1: Recognize the Concept
This problem involves two masses attracting each other via gravity. Since there are no external forces, the center of mass (C.M.) of the system remains stationary.
Step 2: Set Up the System
Let the mass of the smaller sphere be $M$ with radius $R$, and the larger sphere be $5M$ with radius $2R$. Their initial separation between centers is $12R$.
They move towards each other until they collide. Because the radii are $R$ and $2R$, when they collide, the centers are separated by $3R$. Thus, the total distance they collectively cover is:
$$
\text{Total distance covered} = (12R - 3R) = 9R.
$$
Step 3: Express the Movement Using the Center of Mass Condition
Let:
$x_1$ = distance covered by the smaller sphere (mass $M$)
$x_2$ = distance covered by the larger sphere (mass $5M$)
Since the center of mass remains fixed, we have:
$$
M \times x_1 = 5M \times x_2 \quad \Rightarrow \quad x_1 = 5\,x_2.
$$
Additionally, the sum of the distances covered by both spheres before collision is $9R$:
$$
x_1 + x_2 = 9R.
$$
Step 4: Solve for $x_1$ (Distance Covered by Smaller Sphere)
Using $x_1 = 5\,x_2$ and $x_1 + x_2 = 9R$:
\[
5\,x_2 + x_2 = 9R \quad \Rightarrow \quad 6\,x_2 = 9R \quad \Rightarrow \quad x_2 = \frac{9R}{6} = 1.5R.
\]
\[
\text{Hence} \quad x_1 = 5\,x_2 = 5 \times 1.5R = 7.5R.
\]
Step 5: State the Final Answer
The distance covered by the smaller sphere before they collide is $7.5R$.
