Why fuel is mass: the rocket equation, explained
Published
In 1903 Konstantin Tsiolkovsky published the equation that every rocket still lives by[2]. Its lesson fits in four words: fuel is mass too. Every kilogram of fuel you carry for later has to be pushed now.
The equation
NASA writes it like this[1]:
Δu = V_eq · ln(m_f / m_e)
- Δu is the change in speed the rocket can make.
- V_eq is how fast the exhaust leaves the engine.
- m_f is the full mass at the start: the rocket plus all its propellant[1].
- m_e is the empty mass at the end.
The logarithm is the whole story. To double the change in speed, you don’t double the fuel: you square the ratio of full to empty mass. Each extra km/s costs more than the last, because the fuel for it also has to carry the fuel for everything after it.
What helps
Only two things move the numbers:
- A faster exhaust. Speed change grows in step with exhaust speed. The Project Daedalus study was designed around exhaust speeds of 10,600 and 9,210 km/s[3].
- Less to carry. A lighter ship, or a fuller tank relative to the ship, buys more change in speed.
In game
Hostile Flagship uses the equation for every jump, turned round to give the fuel burned:
fuel = mass × (1 − e^(−Δv / v_e))
- In game the drive’s exhaust speed v_e is 10,000 km/s, in line with the Daedalus figures[3].
- Every trip is a 1 g burn: speed up to halfway, flip, brake. So a trip of distance d needs Δv = 2√(d · 9.81 m/s²).
- A route over several links adds up its links’ Δv, and the game picks the route with the least.
For the carrier in game, about 100,000 t:
| Trip | Δv | Fuel | Tanks of 10,000 m³ |
|---|---|---|---|
| The Moon | about 120 km/s | about 1,200 m³ | about 0.12 |
| Mars, at its closest | about 1,500 km/s | about 16,000 m³ | about 1.6 |
| Pluto | about 15,000 km/s | about 360,000 m³ | about 36 |
A flat price per hour with the same Moon cost would make Pluto only about 11 tanks. The exponential is what makes far trips punishing.
The same drive on a small ship goes a long way: the 18.4 t Hawkeye gets about 3,200 km/s of Δv from a 7 m³ tank in game.
Because a ship is lighter after it burns, the way back is cheaper than the way out. The game works this out exactly and warns you when a jump would leave too little to return. More in Jump drives and fuel.
Related
- 36 tanks to Pluto: jump fuel from the rocket equation
- 326,400 km out: the Earth-Moon L1 point
- 350 m long, 80,000 hull: the carrier
Sources
- Ideal Rocket Equation, NASA Glenn Research Center, Beginner's Guide to Aeronautics.
Initially, the full mass of the rocket mf contains the empty mass and all of the propellant at lift off.
- Tsiolkovsky rocket equation, Wikipedia.
The equation is named after—and usually credited to—Konstantin Tsiolkovsky, who derived and published the formula in 1903.
- Project Daedalus, Wikipedia.
producing exhaust velocities of 10,600 km/s and 9,210 km/s respectively.