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

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