Calculate the apparent speed, in km s−1, at which the galaxy is moving with respect to us.?

Question

A certain spectral line of hydrogen gas has a wavelength of 434.2 nm when measured in the laboratory. In the spectrum of a particular distant galaxy, this same spectral line has a wavelength of 455.9 nm.

In order to demonstrate the learning outcomes fully in this question you must show all your working throughout.

(You may assume that the speed of light is 3.0 × 105 km s−1.) Express your answer in scientific notation to an appropriate number of significant figures.

Answer 1

lamda1/lamda0 = sqrt((1+v/c)/(1-v/c)) ==> β (=v/c) = 2/(1+(lamda1/lamda0)^2)-1 ==>

β = -4.87296039080835E-02 ; v = βc = -14618881.172425 m/s

Derivations in the refs. below.