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Space oddity radio

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I'm in a spaceship very far away from earth but traveling straight towards earth with a speed v. It's boring out here, so I decide to try and tune in to some of my favorite earthly radio stations. I remember that my favorite station has a frequency of 100.3 MHz and so tune my radio to exactly this frequency. Amazingly, I hear the radio station just like I do on earth! How fast is my spaceship going in m/s? (Hint: it's not that fast...I think I should check whether my engines are on).

Details and assumptions

  • Photons of electromagnetic radiation have an intrinsic kinetic energy related to their frequency by \(E=hf\) where \(h\) is Planck's constant.
  • The gravitational interaction between photons and earth can be treated via usual Newtonian gravity and \(E=mc^2\) to convert between energy and mass.
  • The total energy of the photons is conserved.
  • The earth can be treated as a sphere of radius 6370 km and mass \(6 \times 10^{24}~kg\). You can ignore rotation of the earth.
  • The speed of light is \(3 \times 10^8~m/s\).
  • Assume the earth is at rest.
3 tries left ·
* Use decimal answers like 3.001 or 6E-10 for 6×10-10. Enter answers to three significant figures – you will not be penalized for too many significant figures.

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