A nucleus disintegrated into two nucleus which have their velocities in the ratio of `2 : 1 ` . The ratio of their nuclear sizes will be

Options

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

Answer

To solve the problem of determining the ratio of the sizes of two nuclei resulting from the disintegration of a nucleus, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We have a nucleus that disintegrates into two smaller nuclei. The velocities of these two nuclei are in the ratio of 2:1.

2. **Apply Conservation of Momentum**: Since the initial momentum of the system is zero (the nucleus is at rest before disintegration), the final momentum must also be zero. Let the masses of the two nuclei be \( m_1 \) and \( m_2 \), and their velocities be \( v_1 \) and \( v_2 \) respectively.

The conservation of momentum can be expressed as:
\[ m_1 v_1 - m_2 v_2 = 0 \]
This implies:
\[ m_1 v_1 = m_2 v_2 \]

3. **Express the Velocity Ratio**: Given that the velocities are in the ratio \( v_1 : v_2 = 2 : 1 \), we can write:
\[ v_1 = 2k \quad \text{and} \quad v_2 = k \]
for some constant \( k \).

4. **Substitute the Velocity into the Momentum Equation**: Substituting the expressions for \( v_1 \) and \( v_2 \) into the momentum equation gives:
\[ m_1 (2k) = m_2 (k) \]
Simplifying this, we find:
\[ 2m_1 = m_2 \quad \Rightarrow \quad \frac{m_1}{m_2} = \frac{1}{2} \]

5. **Relate Mass to Volume**: The mass of a nucleus can be related to its volume and density. Assuming the density (\( \rho \)) is the same for both nuclei, we have:
\[ m_1 = \rho V_1 \quad \text{and} \quad m_2 = \rho V_2 \]
where \( V_1 \) and \( V_2 \) are the volumes of the two nuclei.

6. **Express Volume in Terms of Radius**: The volume of a sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \]
Therefore, we can write:
\[ V_1 = \frac{4}{3} \pi r_1^3 \quad \text{and} \quad V_2 = \frac{4}{3} \pi r_2^3 \]

7. **Set Up the Mass Ratio Using Volume**: Substituting the volumes into the mass ratio gives:
\[ \frac{m_1}{m_2} = \frac{V_1}{V_2} = \frac{r_1^3}{r_2^3} \]
From our earlier result, we have:
\[ \frac{1}{2} = \frac{r_1^3}{r_2^3} \]

8. **Find the Ratio of Radii**: Taking the cube root of both sides, we find:
\[ \frac{r_1}{r_2} = \left(\frac{1}{2}\right)^{1/3} \]

### Final Answer:
The ratio of the nuclear sizes (radii) is:
\[ \frac{r_1}{r_2} = \frac{1}{2^{1/3}} \]