Two spaceships are moving towards each other at a speed of 0.5c relative to the galaxy. Spaceship A sends an emergency signal to spaceship B with a frequency f = 20GHz. What is the frequency of the signal detected by spaceship B?
Relativistic Doppler effect describes how the frequency of waves changes due to the relative motion of the source and observer. Using relativistic doppler effect,
\[ \frac{f_0}{f_s} = \sqrt{\frac{1 + \beta}{1 – \beta}} \]\[ \beta = \frac{v_{\text{rel}}}{c} \]\[ v_{\text{rel}} = \frac{v_1 + v_2}{1 + \frac{v_1 v_2}{c^2}} \]\[v_1 = v_2 = 0.5c\]\[ \Rightarrow v_{\text{rel}} = \frac{0.5c + 0.5c}{1 + \frac{(0.5c)(0.5c)}{c^2}} = \frac{4c}{5}\]\[ \Rightarrow \beta = \frac{v_{\text{rel}}}{c} = \frac{\frac{4c}{5}}{c} = \frac{4}{5} = 0.8\]\(\textrm{Therefore,}\)\[f_0 = 20 \cdot \sqrt{\frac{1 + 0.8}{1 – 0.8}} = 60 \, \text{GHz}\]
