Q172 / 180 · KCET 2015 · Physics
A nucleus at rest splits into two nuclear parts having radii in the ratio \( 1: 2 \). Their velocities are
in the ratio
in the ratio
- Correct answerA8:1
- B4:1
- C6:1
- D\( 2: 1 \)
Solution
Given, nucleus at rest splits into two nuclear parts, then
\[ \begin{aligned} m_{1} v_{1} &=m_{2} v_{2} \\ \Rightarrow \frac{v_{1}}{v_{2}} &=\frac{m_{2}}{m_{1}} \end{aligned} \]
We know that \( A \propto R^{2} \)
Also, \( A \propto m \)
\[ \begin{array}{l} \Rightarrow m \propto R^{3} \\ \Rightarrow \frac{v_{1}}{v_{2}}=\frac{R_{2}{ }^{3}}{R_{1}{ }^{3}} \end{array} \]
Now, given ratio of radii is \( 1: 2 \).
Therefore, the velocities are in the ratio \( 8: 1 \).
\[ \begin{aligned} m_{1} v_{1} &=m_{2} v_{2} \\ \Rightarrow \frac{v_{1}}{v_{2}} &=\frac{m_{2}}{m_{1}} \end{aligned} \]
We know that \( A \propto R^{2} \)
Also, \( A \propto m \)
\[ \begin{array}{l} \Rightarrow m \propto R^{3} \\ \Rightarrow \frac{v_{1}}{v_{2}}=\frac{R_{2}{ }^{3}}{R_{1}{ }^{3}} \end{array} \]
Now, given ratio of radii is \( 1: 2 \).
Therefore, the velocities are in the ratio \( 8: 1 \).
Hence, the correct answer is option: 'A'.