Infinite fuel. Finite speed.

What you're seeing

The chart is a shopping question: you have one tonne of payload to speed up, and you are deciding how much propellant to buy. Everything happens in empty space — no gravity, no air, nothing to fight except the arithmetic. The bottom axis is the propellant load, stretched so that each step rightward multiplies the load by ten. The curve is the speed you end up with. And the first thing it does is betray you: it climbs beautifully for the first few tonnes, then bends, then crawls.

The reason is the loop in the problem. The engine speeds up whatever mass is still aboard — and most of that mass is the propellant you have not burned yet. Buy more fuel and you slow down the very burn you bought it for. Work the loop out exactly and the speed gained is the exhaust velocity times the logarithm of the launch-to-empty mass ratio: to add one more fixed slice of speed, you must multiply the rocket. Press double the fuel a few times and read the receipts. There is a wrinkle worth catching: near the start, while the payload still outweighs the fuel, a doubling can buy more than the last — the logarithm hasn't bitten yet (the gain peaks near 2 t here). Past that shoulder it bites forever: every further doubling buys less, without end.

Now look at the dashed line, because it is the part almost nobody expects. A real rocket cannot be all fuel: tanks, engines and plumbing come along, here about a tenth of the propellant they hold. That fixed overhead does something brutal — it caps the mass ratio itself. Push the slider all the way to ten thousand tonnes; try to feel how absurd that is for one tonne of payload. The curve creeps toward the ceiling vₑ ln(1 + 1/s) and never touches it. The proof script drives it to a million tonnes: still underneath, by a thousandth of a kilometre per second. Infinite fuel. Finite speed. No cleverness with the tank's shape fixes this — only a better engine (raise vₑ) or lighter tanks (lower s) move the line at all.

Then press add a second stage. Same total propellant, split into a rocket riding a rocket, and the cyan curve goes through the ceiling. Nothing magical happened mid-flight: partway through, the first stage's empty tank — pure dead mass — is dropped, and the second stage's logarithm starts from a lighter place. The proof pins down exactly why this works with a control that surprises most people: set the structure to zero, and — in this model — staging gains exactly nothing; the two burns' logarithms telescope back into one. Staging is not about restarting an engine. It is about refusing to keep carrying a tank you have already emptied. The page also finds the best split live — and it is wildly lopsided: at 200 t of propellant, the best split puts about 187 t of it in the first stage. Real rockets are first-stage-heavy for related reasons, among many others this model leaves out.

This is why real launchers are towers of nested stages that are 90-odd percent propellant, and why their payload is a sliver in the mass bar under the chart. A real mission to orbit also pays tolls this idealized picture leaves out — climbing against gravity, pushing through air — which only squeeze the sliver further. The tyranny is not engineering timidity. It is a logarithm, and everyone who flies pays it.