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 determining the ratio of the nuclear sizes after a nucleus ruptures into two parts with a velocity ratio of 2:1, we can follow these steps:

### Step 1: Understand the conservation of momentum
When the nucleus ruptures, the total momentum before the rupture must equal the total momentum after the rupture. Since the nucleus is initially at rest, the initial momentum is zero.

### Step 2: Set up the momentum equation
Let the masses of the two parts be \( m_1 \) and \( m_2 \), and their velocities be \( v_1 \) and \( v_2 \) respectively. According to the conservation of momentum:
\[ m_1 v_1 = m_2 v_2 \]

### Step 3: Use the given velocity ratio
We are given that the velocity ratio \( v_1 : v_2 = 2 : 1 \). This can be expressed as:
\[ \frac{v_1}{v_2} = 2 \quad \Rightarrow \quad v_1 = 2v_2 \]

### Step 4: Substitute the velocity ratio into the momentum equation
Substituting \( v_1 = 2v_2 \) into the momentum equation gives:
\[ m_1 (2v_2) = m_2 v_2 \]
Dividing both sides by \( v_2 \) (assuming \( v_2 \neq 0 \)):
\[ 2m_1 = m_2 \quad \Rightarrow \quad \frac{m_1}{m_2} = \frac{1}{2} \]

### Step 5: Relate mass to nuclear size
The radius of a nucleus is related to its mass number \( A \) (which is proportional to mass) by the formula:
\[ R = R_0 A^{1/3} \]
where \( R_0 \) is a constant.

### Step 6: Express the radii of the two parts
Let \( R_1 \) and \( R_2 \) be the radii of the two parts corresponding to masses \( m_1 \) and \( m_2 \):
\[ R_1 = R_0 m_1^{1/3} \quad \text{and} \quad R_2 = R_0 m_2^{1/3} \]

### Step 7: Find the ratio of the radii
Now, we can find the ratio of the radii:
\[ \frac{R_1}{R_2} = \frac{R_0 m_1^{1/3}}{R_0 m_2^{1/3}} = \frac{m_1^{1/3}}{m_2^{1/3}} \]
Substituting \( \frac{m_1}{m_2} = \frac{1}{2} \):
\[ \frac{R_1}{R_2} = \left(\frac{1}{2}\right)^{1/3} = \frac{1}{2^{1/3}} \]

### Final Answer
Thus, the ratio of the nuclear sizes is:
\[ \frac{R_1}{R_2} = \frac{1}{2^{1/3}} \]