A nucleus splits into two nuclear parts which have their velocity ratio equal to 5 : 1 . What will be the ratio of their nuclear radius ?

Options

A. `5^(1//3) : 1`
B. `1 : 5^(1//3)`
C. `3^(1//2) : 1`
D. `1 : 3^(1//2)`

Explanation

To solve the problem of finding the ratio of the nuclear radii of two nuclear parts that split from a nucleus with a velocity ratio of 5:1, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem:**
We have a nucleus that splits into two parts with a velocity ratio of \( V_1 : V_2 = 5 : 1 \). We need to find the ratio of their nuclear radii.

2. **Conservation of Momentum:**
Since there are no external forces acting on the system, we can apply the principle of conservation of linear momentum. Initially, the momentum is zero (the nucleus is at rest), so the final momentum must also be zero:
\[ A_1 V_1 + A_2 V_2 = 0 \]
Here, \( A_1 \) and \( A_2 \) are the masses of the two parts, and \( V_1 \) and \( V_2 \) are their respective velocities.

3. **Expressing Velocities:**
Rearranging the momentum equation gives:
\[ A_1 V_1 = -A_2 V_2 \]
Taking magnitudes, we have:
\[ A_1 V_1 = A_2 V_2 \]

4. **Substituting the Velocity Ratio:**
Given the velocity ratio \( V_1 : V_2 = 5 : 1 \), we can express \( V_2 \) in terms of \( V_1 \):
\[ V_2 = \frac{1}{5} V_1 \]
Substituting this into the momentum equation:
\[ A_1 V_1 = A_2 \left(\frac{1}{5} V_1\right) \]
Simplifying gives:
\[ A_1 = \frac{1}{5} A_2 \]
Thus, we can express the mass ratio as:
\[ \frac{A_1}{A_2} = \frac{1}{5} \]

5. **Relating Mass to Nuclear Radius:**
The nuclear radius \( R \) is related to the mass number \( A \) by the formula:
\[ R \propto A^{1/3} \]
Therefore, we can express the ratio of the radii as:
\[ \frac{R_1}{R_2} = \left(\frac{A_1}{A_2}\right)^{1/3} \]

6. **Calculating the Radius Ratio:**
Substituting the mass ratio into the radius ratio gives:
\[ \frac{R_1}{R_2} = \left(\frac{1}{5}\right)^{1/3} \]

7. **Final Result:**
Thus, the ratio of the nuclear radii is:
\[ \frac{R_1}{R_2} = \frac{1}{5^{1/3}} \]

### Summary:
The ratio of the nuclear radii \( R_1 : R_2 \) is \( 1 : 5^{1/3} \).

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