Question:
With what speed should a galaxy move outward with respect to earth so that the sodium-D line at wavelength $5890 A$ is observed at $5896 A$ ?
Correct Option: 1,
Solution:
$f=f_{0} \sqrt{\frac{1+\beta}{1-\beta}} \quad \beta=\frac{v}{c}$ $\beta=\frac{v}{c}$
$\frac{f}{f_{0}} \sqrt{\frac{1+\beta}{1-\beta}}$
$\left(1+\frac{\Delta f}{f_{0}}\right)^{2}=(1+\beta)(1-\beta)^{-1}$
$\beta$ is small compared to 1
$\left(1+\frac{2 \Delta f}{f_{0}}\right)=(1+2 \beta)$
$\beta=\frac{\Delta f}{f_{0}}=\frac{v}{c}$
$\mathrm{v}=6 \times \frac{\mathrm{c}}{5890}=305.6 \mathrm{~km} / \mathrm{s}$