Rocket Equation Calculator

The Tsiolkovsky equation follows directly from conservation of momentum for a vehicle that expels mass at a fixed speed. Over an instant, m dv equals minus the exhaust velocity times dm, and integrating from wet mass to dry mass produces the natural logarithm of the mass ratio. The logarithm is the whole story of rocketry: delta-v grows only as the log of propellant loading, so each additional kilometer per second costs exponentially more propellant. Exhaust velocity is usually quoted through specific impulse, with ve equal to Isp times the standard gravity constant 9.80665 m/s squared. Representative values are about 282 s at sea level and 311 s in vacuum for kerosene and liquid oxygen, roughly 452 s for the hydrogen-oxygen RS-25 giving ve near 4,430 m/s, 320 s for storable hypergolics, 250 to 270 s for solid motors, and 1,500 to 4,000 s for electric propulsion. Mission planning compares the result against a delta-v budget. Low Earth orbit needs about 9.3 to 9.5 km/s, of which only about 7.8 km/s is the orbital speed itself and the remainder is gravity and drag losses plus steering. Geostationary transfer adds roughly 2.44 km/s, circularization at GEO about 1.47 km/s, and translunar injection about 3.2 km/s from LEO. With ve of 3,000 m/s, reaching LEO in one stage would require a mass ratio near 23, which is why staging exists: each stage is solved separately and the delta-v values are summed. The equation itself assumes constant exhaust velocity, no external forces, and no atmosphere, so gravity and drag losses must be added by hand.

Konstantin Tsiolkovsky, a largely self-taught schoolteacher in Kaluga, derived the relation in 1897 and published it in 1903 in the journal Nauchnoye Obozreniye under the title Exploration of Outer Space by Means of Reaction Devices. The same mathematics had actually appeared earlier in obscurity: William Moore of the Royal Military Academy at Woolwich published a treatise on the motion of rockets in 1813 that contains the equivalent result, developed for military black-powder rockets rather than spaceflight. Tsiolkovsky went further than the algebra, arguing for liquid oxygen and liquid hydrogen propellants and, in 1929, for multistage rocket trains as the only practical route to orbit. The equation was independently rediscovered by the other founding theorists of astronautics. Robert Goddard, whose 1919 Smithsonian paper A Method of Reaching Extreme Altitudes laid out the same scaling, patented multistage and liquid-fuel concepts in 1914 and flew the first liquid-propellant rocket on 16 March 1926. Hermann Oberth published Die Rakete zu den Planetenraeumen in 1923, which directly inspired the German rocket societies and, through Wernher von Braun, the V-2 and later the Saturn V. Yuri Kondratyuk independently worked out both the equation and the lunar orbit rendezvous approach around 1919. Every launch vehicle since has been sized by the same logarithm.