A nucleus disintegrates into two smaller parts, which have their...

Given here, $\dfrac{{v}_{1}}{{v}_{2}} = \dfrac{3}{2}$

Using conservation of momentum during disintegration, we have

${m}_{1}{v}_{1} = {m}_{2}{v}_{2} ⇒ \dfrac{{m}_{1}}{{m}_{2}} = \dfrac{2}{3}$

Since, $m = \dfrac{4}{3}\pi{R}^{3}ρ$

We have $m ∝ {R}^{3}\left(∵ \text{nuclear mass density}, ρ = \text{constant}\right)$

$\dfrac{{m}_{1}}{{m}_{2}} = {\left(\dfrac{{R}_{1}}{{R}_{2}}\right)}^{3}$

$⇒ \dfrac{{R}_{1}}{{R}_{2}} = \left(\dfrac{2}{3}\right)\dfrac{1}{3}$

So, $x = 2$.