A nucleus ruptures into two nuclear parts, which have their velocity ratio equal to `2 : 1`. What will be the ratio of their nuclear size (nuclear radius)?

Options

A. `2^(1//3) : 1`
B. `1 : 2^(1//3)`
C. `3^(1//2) : 1`
D. `1 : 3^(1//2)`

Answer

To solve the problem of finding the ratio of the nuclear sizes (nuclear radii) of two parts resulting from the rupture of a nucleus, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have a nucleus that ruptures into two parts with a velocity ratio of 2:1. We need to find the ratio of their nuclear sizes (radii).

2. **Define Variables**:
Let:
- \( M_1 \) = mass of the first part
- \( M_2 \) = mass of the second part
- \( V_1 \) = velocity of the first part
- \( V_2 \) = velocity of the second part

3. **Conservation of Momentum**:
Since the initial momentum of the nucleus is zero (it is at rest), the final momentum must also be zero. Therefore, we can write:
\[ M_1 V_1 + M_2 V_2 = 0 \]
Rearranging gives:
\[ M_1 V_1 = -M_2 V_2 \]

4. **Finding the Velocity Ratio**:
From the equation above, we can express the ratio of the velocities:
\[ \frac{V_1}{V_2} = -\frac{M_2}{M_1} \]
Given that the velocity ratio is \( \frac{V_1}{V_2} = \frac{2}{1} \), we can equate:
\[ \frac{2}{1} = -\frac{M_2}{M_1} \]
This implies:
\[ \frac{M_2}{M_1} = 2 \]

5. **Relating Mass to Radius**:
The mass of a nucleus can be expressed in terms of its radius and density:
\[ M = \frac{4}{3} \pi R^3 \rho \]
Since both parts are made of the same material, their densities (\( \rho \)) are equal. Thus, we can write:
\[ \frac{M_2}{M_1} = \frac{R_2^3}{R_1^3} \]

6. **Substituting the Mass Ratio**:
Substituting \( \frac{M_2}{M_1} = 2 \) into the equation gives:
\[ 2 = \frac{R_2^3}{R_1^3} \]

7. **Finding the Radius Ratio**:
Taking the cube root of both sides, we find:
\[ \frac{R_2}{R_1} = 2^{1/3} \]

8. **Final Ratio**:
Therefore, the ratio of the nuclear sizes (radii) is:
\[ \frac{R_1}{R_2} = \frac{1}{2^{1/3}} \]

### Conclusion:
The ratio of the nuclear sizes (radii) of the two parts is \( \frac{R_1}{R_2} = \frac{1}{2^{1/3}} \).