Doppler-effect problem

Question

Please help.

I found this interesting problem on the internet, but I don't know how to start it:

These are the questions:

At what frequency do astronauts hear the hum?

What frequency should the radio receiver be set to?

The problem:

A broadcast is being received in a spaceship equipped with a sensitive radio, which is moving away from a planet at 70% of the speed of light. This broadcast is a 220 Hz hum with AM modulation. The broadcast is by the 600 kHz medium wave radio station, operating on the previously mentioned planet.

I started with the Doppler-effect formula but don't know how to continue.

I appreciate any help you can provide.

Answer 1

You are correct that this an application of the Doppler Effect. Since the speed of the object is close to the speed of light, we have to include special relativity. Furthermore, since the object is moving directly away from the source, we have the Longitudinal Relativistic Doppler Effect. The equation is as follows (from the ever helpful wikipedia).

Source Frequency / Receiver Frequency = sqrt( (1 + β) / (1 - β ) )

where β = Relative speed between the objects / speed of light, which is 0.7 in this case

The source is the planet, and the receiver is the astronaut. Solving for the receiver frequency, we have:

Receiver Frequency = sqrt( (1 - β) / (1 + β ) ) • Source Frequency

Since the objects are moving apart from each other, the frequency shift has to be less that one. This means that our β will take on a positive value in the above equation.

sqrt( (1 - 0.7) / (1 + 0.7) ) = sqrt ( 0.3 / 1.7 ) = 0.42

Receiver Frequency = 0.42 • Source Frequency:

Therefore,

Receiver Hum Frequency = 0.42 • 220 Hz = 92.4 Hz

Receiver Broadcast Frequency = 0.42 •600 kHz = 252 kHz