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Question

(B) The problem involves two stages of the Doppler effect for light.First,the moon acts as an observer receiving the signal from the rocket moving at

Vedclass · Accepted Answer

$\frac{c}{c - 2v}f$

Vedclass · Answer

(B) The problem involves two stages of the Doppler effect for light.First,the moon acts as an observer receiving the signal from the rocket moving at speed $v$. The frequency received by the moon is $f_1 = f \left( \frac{c}{c - v} \right)$.Second,the moon acts as a source reflecting this signal back to the rocket,which is moving towards the moon at speed $v$. The frequency received by the astronaut is $f' = f_1 \left( \frac{c + v}{c} \right)$.Substituting $f_1$,we get $f' = f \left( \frac{c}{c - v} \right) \left( \frac{c + v}{c} \right) = f \left( \frac{c + v}{c - v} \right)$.Using the binomial approximation for $v Alternatively,using the formula $f' = f \left( \frac{c + v}{c - v} \right) = f \left( \frac{c - v + 2v}{c - v} \right) = f \left( 1 + \frac{2v}{c - v} \right) \approx f \left( 1 + \frac{2v}{c} \right) = f \left( \frac{c + 2v}{c} \right)$.However,in the limit $v << c$,the expression simplifies to $f' = f \left( \frac{c}{c - 2v} \right)$.

$A$ rocket is moving towards the moon with a speed $v$. The astronaut in the rocket sends signals of frequency $f$ towards the moon and receives them back after reflection from the moon. What will be the frequency of the signal received by the astronaut? (Assume $v << c$)

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