Animated Solution for Physics - Waves: In hydrogen spectrum, the wavelength of $H_{\alpha }$ line is $656\text{ nm}$; whereas in the spectrum of a distant galaxy $H_{\alpha }$ line wavelength is $706\text{ nm}$. Estimated speed of galaxy with respect to earth is

Imagine you are standing on the side of a highway, listening to the siren of an ambulance as it speeds past you. You notice the pitch of the siren drops the moment it passes. This is the classic Doppler effect for sound. Now, imagine replacing that ambulance with an entire galaxy, and the sound waves with light waves.

When a distant galaxy moves away from us, the light it emits gets stretched out. In the language of physics, its wavelength increases. Because red light has a longer wavelength than blue light, this stretching is called a redshift. In our problem, the famous $H_{\alpha }$ line of hydrogen, which normally has a wavelength of $656\text{ nm}$, is observed at $706\text{ nm}$. This tells us immediately that the galaxy is receding from Earth.

For sound, the Doppler effect depends on whether the source or the observer is moving relative to the medium (like air). But light doesn't need a medium, and according to Einstein's theory of relativity, the speed of light $c$ is constant for all observers. Therefore, we must use the relativistic Doppler effect formula.

For a source moving away from the observer with velocity $v$, the observed wavelength ${\lambda }^{\prime }$ is given by:

This elegant equation accounts for both the relative motion and the time dilation experienced by the moving galaxy.

Let's substitute our known values into the master equation. We have ${\lambda }^{\prime }=706\text{ nm}$ and $\lambda =656\text{ nm}$:

To liberate $v$ from the square root, we square both sides. The ratio $\frac{706}{656}$ is approximately $1.0762$, and squaring it gives us about $1.16$:

Now, it's just a matter of simple algebra. Cross-multiplying gives:

Bringing the $v$ terms to the left and the $c$ terms to the right:

Finally, we isolate $v$:

Knowing that the speed of light $c$ is $3\times 10^8\text{ m/s}$, we plug it in:

And there we have it! The galaxy is zooming away from us at a staggering $2.2\times 10^7\text{ m/s}$. The math perfectly aligns with our physical intuition, revealing the dynamic and ever-expanding nature of our universe.