A heavy nucleus at rest breaks into two fragments which fly off with velocities in the ratio `8 : 1`. The ratio of radii of the fragments is.

Options

A. `1 : 2`
B. `1 : 4`
C. `4 : 1`
D. `2 : 1`

Explanation

To solve the problem step by step, we will use the conservation of momentum and the relationship between the mass number and the radius of the fragments.

### Step-by-Step Solution:

1. **Understanding the Problem**:
A heavy nucleus at rest breaks into two fragments. The velocities of the fragments are given in the ratio of 8:1. We need to find the ratio of the radii of these fragments.

2. **Applying Conservation of Momentum**:
Since the nucleus is initially at rest, the total momentum before the decay is zero. Therefore, the momentum after the decay must also be zero. If we denote the masses of the fragments as \( M_1 \) and \( M_2 \), and their velocities as \( V_1 \) and \( V_2 \) respectively, we can write:
\[ M_1 V_1 + M_2 V_2 = 0 \]
This implies:
\[ M_1 V_1 = -M_2 V_2 \]

3. **Using the Given Velocity Ratio**:
We are given that the velocities are in the ratio:
\[ \frac{V_1}{V_2} = \frac{8}{1} \]
From this, we can express \( V_1 \) in terms of \( V_2 \):
\[ V_1 = 8 V_2 \]

4. **Substituting into the Momentum Equation**:
Substituting \( V_1 \) into the momentum equation gives:
\[ M_1 (8 V_2) = M_2 V_2 \]
Dividing both sides by \( V_2 \) (assuming \( V_2 \neq 0 \)):
\[ 8 M_1 = M_2 \]
This implies:
\[ \frac{M_2}{M_1} = 8 \]

5. **Relating Mass Number to Radius**:
The radius \( R \) of a nucleus is related to its mass number \( A \) by the formula:
\[ R \propto A^{1/3} \]
If we denote the mass numbers of the two fragments as \( A_1 \) and \( A_2 \), we can write:
\[ \frac{R_1}{R_2} = \left(\frac{A_1}{A_2}\right)^{1/3} \]

6. **Finding the Mass Numbers**:
Since \( M_1 \) and \( M_2 \) are proportional to \( A_1 \) and \( A_2 \) respectively, we have:
\[ \frac{M_2}{M_1} = \frac{A_2}{A_1} = 8 \]
This means:
\[ A_2 = 8 A_1 \]

7. **Calculating the Ratio of Radii**:
Substituting \( A_2 = 8 A_1 \) into the radius ratio equation gives:
\[ \frac{R_1}{R_2} = \left(\frac{A_1}{8 A_1}\right)^{1/3} = \left(\frac{1}{8}\right)^{1/3} = \frac{1}{2} \]

8. **Conclusion**:
Therefore, the ratio of the radii of the fragments is:
\[ \frac{R_1}{R_2} = \frac{1}{2} \]

### Final Answer:
The ratio of the radii of the fragments is \( \frac{1}{2} \).