The recoil speed ( in m/s) of a hydrogen atom after it emits a photon in going from $n=5$ state to $n=1$ state ,closest to nearest integer

$E=R_{\infty }hc\left(1-\frac{1}{25}\right)$

Momentum of photon emitted is

        $p=\frac{E}{c}=R_{\infty }h\left(\frac{24}{25}\right)$

Recoil momentum of H-atom will also be $p$.

$$\begin{gathered} \Rightarrow mv=p \\ \Rightarrow v=\frac{p}{m}=\frac{\left(1.097\times {10}^7\right)\left(6.626\times {10}^{-34}\right)24}{\left(25\right)\left(1.67\times {10}^{-27}\right)} \\ \Rightarrow v=4.178 m{s}^{-1} \end{gathered}$$