Classical Mechanics
# Waves

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?