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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.

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Recordings that get played speakers can be very detailed, consisting of tens or hundred of distinct sound sources, each of which contribute a continuum of frequency components. Yet there is only one speaker. How is it possible for a single speaker to faithfully reproduce complex sounds?

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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 \(c = 1235~\mbox{km/h}\).

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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.

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**in kilometers** to accomplish this?

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