Question

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

• A. 8:1
• B. 6:1
• C. 4:1
• D. 2:1

Hint:

As the nucleus was initially at rest, hence the net force acting on the nucleus must be zero. 

Answer 1

The Correct Option is A

Given: \(\frac{R_{1}}{R_{2}}=\frac{1}{2}\)
According to the law of conservation of linear momentum, we have

m₁V₁ = m₂V₂
\(\frac{v_{1}}{v_{2}}=\frac{m_{2}}{m_{1}}=\left(\frac{R_{2}}{R_{1}}\right)^{3}\left(\because m \propto A \propto R^{3}\right)\)
i.e. \(\frac{v_{1}}{v_{2}}=\left(\frac{2}{1}\right)^{3}=\frac{8}{1}\)

Therefore, Option A) 8:1 is the correct answer.

Answer 2

As the nucleus was initially at rest, hence the net force acting on the nucleus must be zero. Now, from the law of conservation of linear momentum, if the net external force acting on a body is zero then the momentum of the body will remain unchanged. 

Hence net momentum of the system before and after the event must be zero.

Formula used:

Since \(∑F_{ext}=0\)

So, P_i = P_f

But the initial momentum of the body, P_i = 0  as initially the body was at rest.

\(P_f= m_1\overrightarrow{v_1}+m_2\overrightarrow{v_2}\)

Or \(m_1\overrightarrow{v_1}+m_2\overrightarrow{v_2}=0\)

Complete step-by-step solution:

Given, the ratio of their radii = 1:2

i.e. \(\frac{r_{1}}{r_{2}}=1:2\)

but \(ρ=\frac{m}{V}\)

so, \(m=ρ×V= ρ\times\frac{4}{3}πr^{3}\)

Now, taking the ratio of both the masses so that the density term gets cancelled

 \(\frac{m_1}{m_2}=\frac{ρV_1}{ρV_2}=\frac{V_1}{V_2}=\frac{4πr_13^3}{4πr_23^3}=\frac{r_3^1}{r_3^2}=(\frac{r_1}{r_2})^3=(\frac{1}{2})^3=\frac{1}{8}\)

Now as we require the ratio of velocities \(\frac{v_1}{v_2}\)

Hence, from \(m_1\overrightarrow{v_1}+m_2\overrightarrow{v_2}=0\)

\(m_1\overrightarrow{v_1}=-m_2\overrightarrow{v_2}\)

\(\frac{|v_1|}{|v_2|}=\frac{m_2}{m_1}\) (Neglecting negative sign as we are asked about magnitudes only)

As calculated \(\frac{m_1}{m_2}=\frac{1}{8}\)

so, \(\frac{m_1}{m_2}= \frac{8}{1}\)

hence,  \(\frac{v_1}{v_2}=\frac{8}{1}\)

So, the correct option is A.

A nucleus at rest splits into two nuclear parts having radii in the ratio 1:2. Their velocities are in the ratio 8:1.