A cosmologist who is studying the light from a galaxy has identified the spectru

Question

A cosmologist who is studying the light from a galaxy has identified the spectrum of hydrogen but finds that the wavelengths are somewhat shifted from those found in the laboratory. In the lab, the H line has a wavelength of 656.3 nm. The cosmologist is using a transmission diffraction grating having 5808 lines/cm in the first order and finds that the first bright fringe for the H line occurs at +23.41 degrees from the central spot.

(a) How fast is the galaxy moving? Express your answer (a)in and (b)as a percentage of the speed of light.
[My incorrect answer: 1.5E7 m/s]

(b) Is the galaxy moving towards us or away from us?
ANSWER: Away from us.

Explanation / Answer

The number of lines on the grating is N = 5808 lines/cm The width of each slit is d = 1/N                                        = 1.72*10-4 cm                                        = 1.72*10-6 m The angle made by the first bright fringe from the central bright spot is                                      = 23.41o The condition for the grating is                            dsin = m (1.72*10-6 m)sin23.41o = (1)                                  = 6.834*10-7 m                                     = 683.4 nm The original wavelength of H line is 0 = 656.3 nm Therefore the wavelength shift is given by                               ( - 0)/0 = v/c                              (26.9 nm)/(656.5 nm) = v/c                                0.041 = v/c The speed of the galaxy is v = 0.041c                                           = 1.23*107 m/s As the observed wavelength is more than that of the original, the galaxy is moving away from us                                           = 1.23*107 m/s As the observed wavelength is more than that of the original, the galaxy is moving away from us

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