Waves are disturbances that transport energy without transporting mass. Learn the mechanism underlying sound, deep ocean swells, light, and even the levitation of objects in mid air.

An ambulance car is going with a speed of \( 60~\mbox{km/h}\), while a car is trying to go around it in a circle with a speed \(50~\mbox{km/h}\). If the sound that ambulance car emits has a frequency of \(1~\mbox{kHz}\), which frequency does the driver of the car hear **in Hz** when the ambulance car is in the center of the circle it makes, and the car makes an angle \(\theta = 30^\circ \) with the direction of the car?

**Details and assumptions**

- Speed of sound in the air is \(c = 1235~\mbox{km/h}\).

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.

**in kilometers** to accomplish this?

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