Rocket Equation
Build a rocket, adjust fuel. Builds on Momentum & Impulse.
A rocket in deep space (no gravity, no air) has a mass of 1000 kg including fuel. Its exhaust velocity is 3 km/s. It burns all its fuel (900 kg), leaving just the 100 kg payload.
What's the rocket's final speed?
About 6.9 km/s — more than double the exhaust speed! The rocket equation is Δv = v_e × ln(m₀/m_f) = 3 × ln(1000/100) = 3 × 2.303 ≈ 6.9 km/s. The logarithm is key: you CAN exceed your exhaust velocity, but it takes exponentially more fuel.
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Adjust the fuel fraction and exhaust velocity. Watch the rocket accelerate as it gets lighter — the acceleration INCREASES as fuel is burned!
Δv = vₑ ln(m₀/mf)
To get Δv = 2vₑ, you need m₀/mf = e² ≈ 7.4. So for every 1 kg of payload, you need 6.4 kg of fuel. For Δv = 3vₑ: 19 kg of fuel per kg of payload. It's brutal.
The Saturn V rocket was 85% fuel by mass. It burned 15 tonnes of fuel per SECOND. All that fuel to send just 47 tonnes to the Moon — the tyranny of the rocket equation.
SpaceX recovers and reuses boosters because the rocket equation makes fuel cheap but hardware expensive. Reusability changes the economics, not the physics.
Ion engines have v_e ≈ 30 km/s (vs 3 km/s for chemical). This means a MUCH better mass ratio for the same Δv. The trade-off: extremely low thrust, so they take months to accelerate.
“The rocket equation is beautiful and cruel. It says you can go anywhere in the universe — but the fuel cost grows exponentially with speed. Every space mission is a battle against this logarithm.”
Rocket Builder
Design a rocket: choose exhaust velocity, fuel mass, and payload. See if you can reach orbit, escape Earth, or even get to Mars.
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Try to reach orbit (Δv ≈ 9.5 km/s) with chemical engines (v_e = 3 km/s). You'll need m₀/mf ≈ 24! Now try with ion engines (v_e = 30 km/s) — the ratio drops to just 1.37.
The rocket equation has a logarithm — doubling the fuel doesn't double the speed. It's the most important equation in space exploration, and the main reason spaceflight is so hard.