With what speed should a galaxy move outward with respect to

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 $5896 A$ ?

  1. $306 \mathrm{~km} / \mathrm{sec}$

  2. $322 \mathrm{~km} / \mathrm{sec}$

  3. $296 \mathrm{~km} / \mathrm{sec}$

  4. $336 \mathrm{~km} / \mathrm{sec}$


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

$\left(1+\frac{2 \Delta f}{f_{0}}\right)=(1+2 \beta)$

$\beta=\frac{\Delta f}{f_{0}}=\frac{v}{c}$

$\mathrm{v}=6 \times \frac{\mathrm{c}}{5890}=305.6 \mathrm{~km} / \mathrm{s}$

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