Tsiolkovsky Rocket Equation

The logarithm is the enemy of rocketry. To double your Δv you must square your mass ratio — a rocket that needs ratio 4 for one stage needs ratio 16 for two stages of the same Δv. This is why reaching LEO from the ground (~9.4 km/s total) demands that roughly 85–90% of liftoff mass is propellant, and why staging exists: each discarded empty tank resets the mass ratio for the next burn. The animation below shows how Δv accumulates as propellant burns off. Watch how quickly the tank empties even for a modest Δv.

A rocket has nothing to push against, so it moves by hurling propellant out the back: every kilogram flung rearward at exhaust speed v_e nudges the ship forward (Newton's third law). Add up those nudges as the ship keeps getting lighter and the logarithm appears. The practical lesson is that how fast you throw the exhaust — what specific impulse measures — matters more than how much fuel you pack: doubling v_e doubles your Δv at the very same mass ratio.

Replace Newtonian Δv with rapidity — the relativistic quantity that still adds linearly — and the exact same log-of-mass-ratio form holds at any speed. So a rocket's Δv budget has no ceiling even though its speed can never reach c. The catch is brutal: a matter–antimatter photon rocket (v_e = c) reaching 0.9c needs a mass ratio near 4.4, and a round trip that also brakes at each end raises that to the fourth power — over 350.