# Space oddity radio

**Classical Mechanics**Level 4

*105 points*

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.

**Your answer seems reasonable.**

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