$A$ rocket is moving towards the moon with a speed $v$. The astronaut in the rocket sends signals of frequency $f$ towards the moon and receives them back after reflection from the moon. What will be the frequency of the signal received by the astronaut? (Assume $v << c$)

If the velocity of a galaxy relative to the Earth is $1.2 \times 10^6 \, m/s$,then the percentage increase in the wavelength of light from the galaxy as compared to a similar source on Earth will be ...............$\%$

$A$ star is moving towards the earth with a speed of $4.5 \times 10^6 \ m/s$. If the true wavelength of a certain line in the spectrum received from the star is $5890 \ \mathring A$,its apparent wavelength will be about........$\mathring A$ $[c = 3 \times 10^8 \ m/s]$

With what speed should a galaxy move outward with respect to Earth so that the sodium-$D$ line at wavelength $5890 \ \mathring{A}$ is observed at $5896 \ \mathring{A}$? (in $\text{km/s}$)

Assuming that the universe is expanding,if the spectrum of light coming from a star which is moving away from the Earth is tested,then in the wavelength of light:

For sound waves,the Doppler formula for frequency shift differs slightly between the two situations:
$(i)$ source at rest; observer moving,and
$(ii)$ source moving; observer at rest.
The exact Doppler formulas for the case of light waves in vacuum are,however,strictly identical for these situations. Explain why this should be so. Would you expect the formulas to be strictly identical for the two situations in case of light travelling in a medium?