Calculate the wavelength problem

[NAkid]

Homework Statement

The wavelength of a particular ultraviolet radiation from hydrogen atoms is 1.216 e-7 m. Calculate the wavelength when the corresponding radiation from hydrogen in a distant galaxy is observed here on earth. The galaxy is receding at a speed of 0.11 c.

Homework Equations

The Attempt at a Solution

I'm not sure what to do with the "receding at a speed of .11c" to solve this problem.. suggestions?

 

[Shooting Star]

When there is relative motion between the light or EM source and the observer, and the relative speed is high, the relativistic Doppler effect formula has to be used. The simplest case is when the motion is directly away from each other with a relative speed of v. If the frequency of the source in its rest frame is f, then the the observer will measure the frequency of the radiation as f' = f*sqrt[(c-v)/(c+v)].

Receding at a speed of .11c means that the relative velocity is 0.11c away from each other.

(To be very correct, the galaxies show redshift because of the expansion of the Universe, but the special relativistic formula will suffice for a relative speed of 0.11c.)

 

[Moetasim]

I would approach this problem by using the formula for Doppler shift, which takes into account the motion of the source of the radiation. In this case, the wavelength of the radiation from the distant galaxy can be calculated using the formula:

λ' = λ (1 + v/c)

Where λ' is the observed wavelength, λ is the original wavelength, v is the speed of the source (in this case, the galaxy receding at 0.11c), and c is the speed of light.

Plugging in the given values, we get:

λ' = (1.216 e-7 m) (1 + 0.11) = 1.35 e-7 m

Therefore, the observed wavelength of the radiation from the distant galaxy would be 1.35 e-7 m on Earth. This is longer than the original wavelength due to the Doppler effect, which causes the wavelength to stretch as the source moves away from the observer.

It is also worth noting that this calculation assumes the galaxy is moving directly away from Earth. If the motion is at an angle, the formula would need to be modified to account for the component of the velocity that is directed towards or away from the observer.