Question

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?

• A. 306 km/sec
• B. 322 km/sec
• C. 296 km/sec
• D. 336 km/sec

Answer 1

The correct option is A

306 km/sec

Step 1: Speed and Wavelength.

1. Speed- The pace at which an object's location changes in any direction.
2. Wavelength- The wavelength is the distance between successive crests or troughs.
3. We know that speed of light is $3\times {10}^{8}m/s$

Step 2: Calculations

$\begin{array}{l}f=f_0=\sqrt{\frac{1+\beta }{1-\beta }}\end{array}$

$\begin{array}{l}\frac{f}{f_0}=\sqrt{\frac{1+\beta }{1-\beta }}\end{array}$

$\begin{array}{l}[1+\frac{\mathrm{\Delta }f}{f_0}{]}^2=(1+\beta \left)\right(1-\beta {)}^{-1}\end{array}$

$\begin{array}{l}[1+\frac{2\mathrm{\Delta }f}{f_0}{]}^2=(1+2\beta )\end{array}$

$\begin{array}{l}\beta =\frac{\mathrm{\Delta }f}{f_0}=\frac{v}{c}\end{array}$

$\begin{array}{l}v=6\times \frac{c}{5890}=\frac{6\times 3\times {10}^8{\displaystyle \raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right.}}{5890}=0.003056\times {10}^8=305.6\raisebox{1ex}{$km$}\!\left/ \!\raisebox{-1ex}{$s$}\right.\end{array}\approx 306\raisebox{1ex}{$km$}\!\left/ \!\raisebox{-1ex}{$s$}\right.$

Final Answer: The correct option is (a)306 km/sec