Question

A nucleus disintegrates into two nuclear parts, in such a way that ratio of their nuclear sizes is \(1 : 2^{2/3}\) . Their respective speed have a ratio of n : 1. The value of n is _______.

Answer 1

Correct Answer: 2

Given: 

• Ratio of nuclear sizes: \( r_1 : r_2 = 1 : 2^{1/3} \)
• Ratio of speeds: \( V_1 : V_2 = n : 1 \)

Step 1: Apply Conservation of Linear Momentum

The total momentum before disintegration is equal to the total momentum after disintegration. Assuming the nucleus is initially at rest, the total momentum before disintegration is zero. Thus, the momenta of the two fragments after disintegration must be equal and opposite:

\[ m_1 V_1 = m_2 V_2, \]

where:

• \( m_1 \) and \( m_2 \) are the masses of the two fragments
• \( V_1 \) and \( V_2 \) are their respective speeds.

Rearranging for the velocity ratio:

\[ \frac{V_1}{V_2} = \frac{m_2}{m_1}. \]

Step 2: Relate Mass and Radius

Assuming the fragments have the same density, their masses are proportional to the cubes of their radii:

\[ \frac{m_2}{m_1} = \frac{r_2^3}{r_1^3}. \]

We are given \( \frac{r_2}{r_1} = 2^{1/3} \). Substituting this into the mass ratio equation:

\[ \frac{m_2}{m_1} = \left( \frac{r_2}{r_1} \right)^3 = \left( 2^{1/3} \right)^3 = 2. \]

Step 3: Find the Ratio of Velocities

Using the velocity ratio equation from Step 1:

\[ \frac{V_1}{V_2} = \frac{m_2}{m_1} = 2. \]

Therefore, \( V_1 : V_2 = n : 1 \), where \( n = 2 \).

Final Answer:

The value of \( n \) is 2.