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.

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?

Worldtech Asked on 25th December 2024 in Astrodynamics.
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    Relativistic Doppler effect describes how the frequency of waves changes due to the relative motion of the source and observer. Using relativistic doppler effect,

    \[ \frac{f_0}{f_s} = \sqrt{\frac{1 + \beta}{1 – \beta}} \]\[ \beta = \frac{v_{\text{rel}}}{c} \]\[ v_{\text{rel}} = \frac{v_1 + v_2}{1 + \frac{v_1 v_2}{c^2}} \]\[v_1 = v_2 = 0.5c\]\[ \Rightarrow  v_{\text{rel}} = \frac{0.5c + 0.5c}{1 + \frac{(0.5c)(0.5c)}{c^2}} = \frac{4c}{5}\]\[ \Rightarrow  \beta = \frac{v_{\text{rel}}}{c} = \frac{\frac{4c}{5}}{c} = \frac{4}{5} = 0.8\]\(\textrm{Therefore,}\)\[f_0 = 20 \cdot \sqrt{\frac{1 + 0.8}{1 – 0.8}} = 60 \, \text{GHz}\]

    techAir Answered on 25th December 2024.
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