A nucleus splits into two nuclear parts having radii in the ratio `1:2` Their velocities are in the ratio

Options

A. `4:1`
B. `8:1`
C. `2:1`
D. `6:1`

Explanation

To solve the problem, we need to determine the ratio of the velocities of two nuclear fragments after a nucleus splits, given that their radii are in the ratio of 1:2.

### Step-by-Step Solution:

1. **Identify the Given Ratios**:
- The radii of the two nuclear parts are given in the ratio \( R_1 : R_2 = 1 : 2 \).

2. **Understand the Relationship Between Mass and Radius**:
- The mass of a nucleus is directly proportional to its volume, which is in turn proportional to the cube of its radius. Therefore, we can express the masses \( m_1 \) and \( m_2 \) in terms of their radii:
\[ m_1 \propto R_1^3 \quad \text{and} \quad m_2 \propto R_2^3 \]

3. **Express the Masses in Terms of the Radii**:
- Let’s denote the masses as:
\[ m_1 = k \cdot R_1^3 \quad \text{and} \quad m_2 = k \cdot R_2^3 \]
- Here, \( k \) is a proportionality constant.

4. **Calculate the Masses Using the Given Ratio**:
- Since \( R_1 : R_2 = 1 : 2 \), we can write:
\[ R_1 = 1x \quad \text{and} \quad R_2 = 2x \]
- Therefore:
\[ m_1 = k \cdot (1x)^3 = k \cdot 1x^3 = kx^3 \]
\[ m_2 = k \cdot (2x)^3 = k \cdot 8x^3 = 8kx^3 \]

5. **Apply Conservation of Linear Momentum**:
- According to the conservation of linear momentum, the total momentum before the split must equal the total momentum after the split:
\[ m_1 v_1 + m_2 v_2 = 0 \]
- Assuming the initial momentum is zero, we can rearrange this to:
\[ m_1 v_1 = -m_2 v_2 \]

6. **Substitute the Masses into the Momentum Equation**:
- Substituting the expressions for \( m_1 \) and \( m_2 \):
\[ (kx^3) v_1 = - (8kx^3) v_2 \]
- We can cancel \( kx^3 \) from both sides (assuming \( k \) and \( x \) are not zero):
\[ v_1 = -8 v_2 \]

7. **Determine the Ratio of Velocities**:
- The ratio of the velocities \( v_1 : v_2 \) is:
\[ v_1 : v_2 = 8 : 1 \]

### Final Answer:
The velocities of the two nuclear parts are in the ratio \( 8 : 1 \).

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