For gas $A , v _{ A }^{2}=\frac{3 kT }{ m }$ $\dots \; (i)$

For a gas molecule, $v ^{2}+ v _{ x }^{2}+ v _{ y }^{2}+ v _{ z }^{2}=3 v _{ x }^{2}\left(\because v _{ x }^{2}= v _{ y }^{2}= v _{ z }^{2}\right)$ or $v _{ x }^{2}=\frac{ v ^{3}}{3}$

From eqn, $(i)$, we get

$w ^{2}= v _{ x }^{2}=\left[\frac{\frac{3 kT }{ m }}{3}\right]=\frac{ kT }{ m }$ $\dots \; (ii)$

For gas $B, v_{B}^{2}=v^{2} \frac{3 k T}{2 m}$ $\dots \; (iii)$

Dividing eqn. $(ii)$ by eqn. $(iii)$, we get

$\frac{ w ^{2}}{ v ^{2}}=\frac{\frac{ kT }{ m }}{\frac{3 kT }{2 m }}=\frac{2}{3}$