ROCKET
Δv = vₑ·ln(m₀/m) — every m/s borrowed from the exhaust, and the ln collects.
Use the simulation above to change the variables and play through the guided stages. The explanation below describes the default starting values; the simulation updates its explanation as you experiment.
Setup
Deep space, nothing to push against, and a machine that is mostly its own fuel. The only way forward is to throw part of yourself backward — and the bookkeeping of that trade is one logarithm.
Principle
Newton’s third law with no ground: hurl mass backward at v_e and the reaction is thrust F = v_eṁ, steady as a metronome. The rocket and its exhaust are a closed system — their total momentum starts at zero and is never allowed to leave.
Solve
Each shed sliver of mass kicks whatever remains, and what remains keeps shrinking — integrate that spiral and out falls Tsiolkovsky’s equation: Δv = v_e ln(m₀/m_dry). The logarithm grows slowly on purpose; astronautics has resented it since 1903.
Burn
The burn, time-compressed and honestly labeled. Thrust never changes, but the ship it pushes keeps losing weight — so the acceleration climbs and the velocity curve bends upward, hardest right before the tank runs dry. The stars stream faster; the camera rides along.
Audit
The ledger: rocket momentum plus exhaust momentum equals zero at every sampled instant — every m/s was borrowed from the plume. And the tyranny, stated plainly: doubling Δv means squaring the mass ratio. That single fact is why rockets stage, and why the sky is expensive.