Rocket equation

The rocket equation, usually called the Tsiolkovsky equation, relates the velocity change Δv a rocket can produce to its exhaust velocity u and the ratio of its initial to final mass. Because the mass ratio enters logarithmically, every increment of additional Δv costs exponentially more propellant — a relation that dominates the economics and engineering of spaceflight.

For chemical rockets, exhaust velocities are limited by the energy density of the propellant to around 4.5 km/s. Reaching low Earth orbit requires roughly 9.4 km/s of Δv, which forces a mass ratio near 8: eight kilograms of propellant per kilogram of everything else. This is why launch vehicles are mostly fuel tanks, why multi-stage rockets exist, and why electric propulsion (with u ≈ 30 km/s) is so attractive for deep-space missions despite its low thrust.

Derived by Konstantin Tsiolkovsky in 1903 in his paper "The Exploration of Cosmic Space by Means of Reaction Devices." Independently rederived by Robert Esnault-Pelterie (1913) and Robert Goddard (1919).