The Rocket Equation
🛰️ Orbital mechanics
Question
Why are rockets 90%+ fuel?
Konstantin Tsiolkovsky's 1903 equation is the entire budget of a rocket: the speed you can add (Δv) equals exhaust velocity times the natural log of the mass ratio. The logarithm is the cruel part: doubling the fuel only buys a little more speed, because the extra fuel also has to be lifted. The only fixes are faster exhaust or dropping dead weight - which is why rockets are staged: burn, throw away the empty tanks, repeat.
The math
Δv = vₑ × ln(m₀/mf) - the natural log of the start-to-end mass ratio (m₀ = full, mf = burnt out). Reaching orbit needs Δv ~ 9.4 km/s (7.7 km/s orbital + ~1.7 km/s gravity and drag losses). With chemical exhaust vₑ ~ 3.5 km/s: m₀/mf = e^(9.4/3.5) = e^2.69 ~ 14. For every 1 kg of payload, ~13 kg of propellant. Saturn V: 2,970 t total for 43.5 t to the Moon (1.5%).
Why it matters
'Rocket science' is literally this equation. It explains why rockets are expensive, why they are staged, and why recovering and reusing the first stage (the biggest, most fuel-guzzling part) is worth billions of dollars.