Tsiolkovsky Rocket Equation

Konstantin Tsiolkovsky published this in 1903, in a Russian journal, working alone as a deaf provincial schoolteacher in Kaluga. He had no laboratory and no rocket. He had momentum conservation and the calculus, and from those two things he derived the equation that still decides whether a mission is possible: $\Delta v = v_e \ln(m_0/m_f)$. Robert Goddard and Hermann Oberth arrived at it independently within the next two decades. Nobody has improved on it, because there is nothing to improve — it is Newton's third law integrated over a burn, and it is exact.

The word doing all the work is logarithm. Delta-v is not proportional to how much propellant you carry; it is proportional to the logarithm of the mass ratio. Burn away 90% of the vehicle and the mass ratio is 10, and $\ln 10 = 2.303$ — so all that propellant buys just 2.3 exhaust velocities of speed. Push to 95% and the ratio doubles to 20, but $\ln 20 = 3.00$, so doubling the propellant fraction added only 30% more delta-v. Each further increment costs exponentially and returns less. That curve, flattening as it climbs, is the reason spaceflight is hard, and no engineering has ever bent it.

Two ways out, and only two. Raise $v_e$, which sits OUTSIDE the logarithm and pays back every gain in full — this is why the industry chases exhaust velocity so relentlessly, and why hydrogen-oxygen engines exist despite hydrogen being a miserable propellant to store and pump. Or split the vehicle into stages, so each stage runs its own rocket equation on its own small mass ratio and the delta-vs add while the ratios multiply. Two stages of mass ratio 4 deliver what one stage of ratio 16 would, and a ratio of 4 is easy to build while 16 is close to impossible.

Everything above is the IDEAL equation, and a real launch does not get it. It assumes no gravity acting along the flight path and no atmosphere, and a launch from Earth's surface has both. Gravity losses — the delta-v spent simply holding the vehicle up while it climbs — plus drag losses come to roughly 1.5 to 2 km/s on the way to low orbit. Orbital velocity there is about 7.8 km/s, so the budget the rocket equation must actually deliver is closer to 9.4 km/s. Leave those losses out and a design closes on paper that will not close on the pad.