The recoil speed of a hydrogen atom after it emits a photon in going from n = 5 state to n = 5 state will be

$\left(\Delta E\right)$ Releases when photon going from $n=5$ to $n=\Delta E$

$=\left(13.6-0.54\right) \mathrm{eV}=13.06 \mathrm{eV}$

$P_{\mathrm{i}}=P_{\mathrm{f}}$ (By linear momentum conservation)

$0=\frac{h}{\lambda }-Mv=V_{\mathrm{Recoil}}=\frac{h}{\lambda M} ...\left(1\right)$

$\& \Delta E=\frac{hc}{\lambda }=\frac{hc}{\lambda M}\times M\Rightarrow Mc{V}_{\mathrm{Recoil}}$

$V_{\mathrm{Recoil} }=\frac{\Delta E}{Mc}=\frac{13.06\times 1.6\times {10}^{-19}}{1.67\times {10}^{-27}\times 3\times {10}^8}=4.17 \mathrm{m} {\mathrm{s}}^{-1}$