Why one more passenger costs so much.

The Tsiolkovsky rocket equation says the velocity a rocket can gain, Δv, is its exhaust velocity v_e times the natural log of its mass ratio R = m₀/m_f (wet mass over dry mass). Because it's a log, not a straight line, each extra unit of Δv costs proportionally more propellant than the last — this is the "tyranny of the rocket equation": fuel to carry fuel is itself mass, and that mass needs its own fuel to move, compounding exponentially. It's why a small increase in payload (or required speed) can demand a disproportionate increase in tank size. The reference line at 9.4 km/s is the standard estimate for reaching low Earth orbit from the ground — it already bakes in gravity and aerodynamic drag losses on top of the roughly 7.8 km/s of raw orbital velocity needed. The ion drive option shown here represents realistic in-space cruise Δv budgets (like NASA's Dawn spacecraft), not launch capability: its thrust is far too low to lift a rocket off a planet's surface, so it only ever flies after a chemical rocket has already reached orbit. The three presets use those vehicles' real wet and dry masses — the Saturn V's first stage moves a 2,970-tonne stack down to 893 tonnes and still only buys about 3.6 km/s, roughly a third of the way to orbit, which is exactly why it needs two more stages above it. These are ideal Δv figures: a real launch also spends 1–2 km/s fighting gravity and drag, which is already accounted for in the 9.4 km/s reference line.