With what speed should a galaxy move outward with respect to Earth, so that the sodium-D line at wavelength 5890 Å is observed at 5896 Å ?

Given, observed wavelength, ${\mathrm{\lambda }}_{\mathrm{o}}=5896\text{Å}$

Actual wavelength, ${\mathrm{\lambda }}_1=5890\text{Å}$

$\begin{array}{c}\therefore \text{ Change in wavelength, }\mathrm{\Delta \lambda }={\mathrm{\lambda }}_{\mathrm{o}}-{\mathrm{\lambda }}_1=5896-5890\\ =6\text{Å}\end{array}$

Speed of galaxy with respect to Earth is given as

$\begin{array}{c}\mathrm{v}=\mathrm{c}\times \frac{\mathrm{\Delta \lambda }}{{\mathrm{\lambda }}_1}\\ =3\times {10}^8\times \frac{6}{5890}\\ =305.6\times {10}^3\mathrm{m}/\mathrm{s}\simeq 306\mathrm{km}/\mathrm{s}\end{array}$