I've just watched an animation movie where a girl goes from Earth to Moon with a little toy rocket. That made me think: How much fuel do you minimally need to get from Earth to the moon and how fast can it be?

## The Distance

The distance between Earth and moon changes. When they are closest it is called perigee. That is approximately 360,000 kilometers. At apogee, the point at which the moon is farthest from Earth, the distance is approximately 405,000 kilometers. (source).

Let's take 360,000 km for the remaining article.

## Maximum accelaration humans can endure

Let's completely ignore what is currently technically possible. We just assume we have that distance on Earth.

Let's say we take 3G of accelaration. That is what astronauts need to endure during take-off (source).

That is 1G is \(9.8 \frac{m}{s^2}\), that means 3G is \(29.4 \frac{m}{s^2}\).

With that speed, you get from 0 to \(100 \frac{km}{h}\)

where v is the velocity (speed), a is the accelaration, and t is the time.

That means:

You get from 0 to 100km/h (62 miles per hour) in less than a second. That is way faster than any production car (source). The Tesla Model S Plaid is at 2 seconds, Ferrari/Posche/Lamborghini are over 2s.

A 3G accelaration is really freaking fast and super uncomfortable. I guess over a longer time it is also pretty dangerous for your health.

## Travel time

Assuming a 3G accelaration, starting with 0km/h and ending with 0km/h - we don't want to crash into the moon. That means we accelarate half the distance and then need to de-accelarate.

So we check how long it takes us to travel half the distance with 3G accelaration. Then we double that time.

It would still take almost **83 minutes**. That is the absolute fastest time possible.
If you're having less luck with the distance and go at "only" 1G (0 to 100km/h in 2.8 seconds)
it would take 2.5 hours.

Apollo 11 needed 76 hours (source).

## Energy: A hard minimum

For an object of mass \(m\) the energy required to escape Earths gravitational field is \(GMm / r\):

- r is radius of the Earth, nominally 6,371 kilometres (3,959 mi),
- G is the gravitational constant,
- M is the mass of the Earth, M = 5.9736 × 1024 kg

That means every kg (kilogram) needs over \(62 \cdot 10^6\) Joule. Or 17.35 kWh.

Astonishingly little, but it adds up as you need to get a lot of weight up.

The Space Shuttle Columbia weights about 3,600 kg. That means one needs 62.5 MWh to lift it. At least 1900kg of liquid hydrogen. Or as much as 31 German households need in energy per year.

Just to leave Earth. You need also quite a bit to land on the moon.

## Energy: Real numbers

SpaceX fuels their crafts not with liquid hydrogen, but with kerosene, which has a lot more energy per gallon. Thanks to this and other advances, Falcon 9’s first stage uses 39,000 gallons of liquid oxygen and almost 25,000 gallons of kerosene, while the second stage uses 7,300 gallons of liquid oxygen and 4,600 gallons of kerosene. Combined, it makes lean mean 75,900 gallons of fuel.

(source)

- 39,000 gallons = 147,631 L. With 1.141 kg/L that makes 168 metric tons.
- 4,600 gallons = 17,412 L. With 0.8 kg/L that maks about 14 metric tons.

So in total 182 tons of fuel.