When the hydrogen atom emits a photon in going from n = 5 to n = 1 state, its recoil speed is nearly

$\text{ Photon energy }=hf=13.6\left[1-\frac{1}{25}\right]\mathrm{eV}=13\mathrm{eV}$

Photon momentum = momentum of hydrogen atom:

$\Rightarrow p=\frac{hf}{c}\text{ or }mv=\frac{hf}{c}$

$\Rightarrow v=\frac{hf}{mc}=\frac{E}{mc}=\frac{13\times 1.6\times {10}^{-19}}{1.67\times {10}^{-27}\times 3\times {10}^8}=4{\mathrm{ms}}^{-1}$