A nucleus disintegrates into two nuclear parts, in such a way that...

Using the law of Momentum conservation

${m}_{1}{v}_{1} + {m}_{2} ( - {v}_{2} ) = 0$
$⇒ {m}_{1}{v}_{1} = {m}_{2}{v}_{2}$
$⇒ \left(\dfrac{{m}_{1}}{{m}_{2}}\right) = \left(\dfrac{{v}_{2}}{{v}_{1}}\right) ⇒ \dfrac{\dfrac{4\piρ{r}^{3}1}{3}}{\dfrac{4\piρ{r}^{3}2}{3}} = \left(\dfrac{{v}_{2}}{{v}_{1}}\right)$

It is given

$\dfrac{{r}_{1}}{{r}_{2}} = \dfrac{1}{{2}^{\dfrac{1}{3}}}$
$\text{and}{\left(\dfrac{{r}_{1}}{{r}_{2}}\right)}^{3} = \left(\dfrac{{v}_{2}}{{v}_{1}}\right)\text{Therefore}, \left(\dfrac{1}{2}\right) = \left(\dfrac{{v}_{2}}{{v}_{1}}\right)$
$⇒ \dfrac{{v}_{1}}{{v}_{2}} = \dfrac{2}{1}$