A nucleus rupture into two nuclear parts which have their velocity ratio equal to 2 : 1. What will be the ratio of their nuclear size (nuclear radius)?

$\text{ Velocity ratio }\left({\mathrm{v}}_1:{\mathrm{v}}_2\right)=2:1\text{. }$

$\text{ Mass }\left(m\right)\propto \text{ volume }\propto r^3$

$\text{ According to law of conservation of momentum }$

$m_1{v}_1=m_2{v}_2$

$\text{ Therefore }\frac{v_1}{v_2}=\frac{m_2}{m_1}=\frac{r_2^3}{r_1^3}$

$\begin{array}{l}\Rightarrow \frac{r_1}{r_2}={\left(\frac{v_2}{v_1}\right)}^{1/3}={\left(\frac{1}{2}\right)}^{1/3}=\frac{1}{2^{1/3}}\\ \Rightarrow r_1:r_2=1:2^{1/3}.\end{array}$