Question

With what speed should a galaxy move outward with respect to earth so that the sodium$-\text{D}$ line at wavelength $5890\dot{\text{A}}$ is observed at $5896\dot{\text{A}}?$

• A. $306\text{km/sec}$
• B. $322\text{km/sec}$
• C. $296\text{km/sec}$
• D. $336\text{km/sec}$

Answer 1

The correct option is A $306\text{km/sec}$

Given:
${\lambda }_{app}=5890\dot{\text{A}};{\lambda }_0=5896\dot{\text{A}}$

Let $v$ be the velocity of the galaxy and $c$ be the velocity of light in vacuum.

For $v<<c$, we know that

When galaxy moving away from the earth

$\frac{{\lambda }_{app}}{{\lambda }_0}=1-\frac{v}{c}(\text{when},v<<c)$

$\Rightarrow \frac{5890}{5896}=1-\frac{v}{3\times {10}^8}$

$\Rightarrow v=3.0529\times {10}^5\text{m/s}\approx 306\text{km/s}$

Hence, option $\left(\text{A}\right)$ is correct.