Electromagnetic radiation energy from the Sun arrives at the upper atmosphere of our planet at a rate of about 1350 J/(s \(\cdot\) m\(^2\)). Use this information, together with the average radius of the Earth's orbit, to show that the Sun radiates energy at a rate of about \(4 \times 10^{26}\) J/s.
We know from radiometric dating of rocks on Earth (and the Moon and Mars) that our solar system is about 4 billion years old. Let's make a naïve hypothesis (like scientists did in the early 1900s) that the Sun is powered by burning hydrocarbons. What mass of gasoline would be needed to power the Sun at a rate of \(4 \times 10^{26}\) J/s for 4 billion years? Compare to the actual mass of the Sun.
Note: The energy density of hydrocarbon fuels, including gasoline, natural gas, dry logs of wood, chocolate, croissants, gummy bears, etc. etc. is \(\approx\) 40 MJ/kg.
In class we analyzed a gas-powered car driving at 70 mph (30 m/s). There was a flow of energy going into the kinetic energy of the wind trail behind the car, and an additional flow of heat energy warming the environment. Approximately how many gallons of gas does it take to drive a car 100 miles at this speed? Show how you worked it out, remember that you can't use the equation you derived in class as a starting point.