hydrogen atom, when vigorously perturbed, can emit light with a frequency of 6.16 × 1014 Hz. When the same light from hydrogen atoms in a distant galaxy is observed on earth, the frequency is 5.31 × 1014 Hz. Calculate the speed at which the galaxy is receding from the earth (in units of the speed of light, c)
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f = 6.16 x 10^14 Hz at rest
fg = 5.31 x 10^14 Hz in distant galaxy
conventional Doppler effect
(f-fg)/f = vg/c where vg is the galaxy recession velocity
vg = 0.138c, this answer is too close to c so the relativistic Doppler effect must be used
fg/f = √((1-b)/(1 +b) where b = vg/c
0.86 = √((1-b)/(1 +b)
0.74 = (1-b)/(1+b)
0.74 + 0.74b = 1 -b
1.74b = 0.26
b = 0.15
vg = 0.15c receding away from the Earth
fg = 5.31 x 10^14 Hz in distant galaxy
conventional Doppler effect
(f-fg)/f = vg/c where vg is the galaxy recession velocity
vg = 0.138c, this answer is too close to c so the relativistic Doppler effect must be used
fg/f = √((1-b)/(1 +b) where b = vg/c
0.86 = √((1-b)/(1 +b)
0.74 = (1-b)/(1+b)
0.74 + 0.74b = 1 -b
1.74b = 0.26
b = 0.15
vg = 0.15c receding away from the Earth