Lunar Lander & the Fuel Identity
Land the module by hand, and see why fuel burned equals gravity times flight time exactly — so minimum fuel is the same thing as minimum time.
Hold the throttle up to slow down. Touch down at under 2.5 m/s or you break the legs. Keyboard: the up arrow is full throttle.
Fuel burned = gravity × flight time. That is the whole game.
The vertical equation is dv/dt = g − u, with u the thrust. Integrate it over the whole descent and the thrust term is just the fuel, so
fuel = g × T − (landing speed)
Start and finish at rest and it collapses to fuel = g × T. Not approximately — exactly, whatever you do with the throttle. Every second in the air costs you g units of fuel whether the engine is lit or not.
| Descent profile | Time | Fuel burned | g × T − v |
|---|---|---|---|
| Fall, then brake at the last moment | 32.0 s | 51.2 | 51.2 |
| Hover 20 seconds first, then brake | 52.0 s | 83.6 | 83.6 |
| Steady 10 m/s all the way down | 785.7 s | 1271.5 | 1271.5 |
| Fall, but never faster than 25 m/s | 32.8 s | 52.9 | 52.9 |
Two columns, four wildly different flight plans, and they agree to better than one part in a billion. Which turns the problem inside out: there is no clever throttle pattern. Minimum fuel is the same thing as minimum time.
So the optimal landing is the terrifying one
Minimum time means falling as fast as you dare and braking as late as you dare — the “suicide burn”. Every gentle, careful, sensible-feeling descent is more expensive than a frightening one, and the gap is not small: from 500 m on the Moon, steady 10 m/s all the way down costs 25× the fuel of a suicide burn. The ratio barely moves with altitude — about 25× from 100 m and from 1,000 m alike.
And hovering is the single most expensive thing a lander can do, because it is pure time with no progress: twenty seconds of hover costs exactly g × 20 = 32.4 units, no matter where you do it or what happens next. That is why real descent profiles look so alarming, and why a pilot who slows down early to “be safe” runs out of fuel and lands hard anyway.
Notes
- Constant mass. The identity assumes the lander does not get lighter as it burns. A real rocket does, which helps you slightly and does not change the shape of the answer.
- Stopping distance is v²/2a, with a the net deceleration after gravity. Brake at exactly that altitude and you arrive at the ground at zero — checked by integration, not by rearranging the formula.
- An engine weaker than gravity can never stop you, at any altitude. The stopping distance is infinite, and no throttle pattern fixes it.
- Free fall from 500 m on the Moon arrives at √(2gh) = 40.2 m/s. On Earth the same drop arrives at 99 m/s, which is why the Moon is the forgiving one despite having no atmosphere at all.
How to use
- Start the descent and use the throttle, or the up arrow, to slow down.
- Watch the dashed line: that is where you must brake at full thrust.
- Touch down under 2.5 m/s or the legs break.
- Try flying it optimally and compare the fuel against your own landing.
Frequently asked questions
How much fuel does a landing need?
Exactly gravity times the total flight time, minus your landing speed. For a soft landing from rest that is simply g times T — not approximately, exactly, and regardless of how you use the throttle. Every second in the air costs g units of fuel whether the engine is lit or not.
What is the most fuel-efficient way to land?
The fastest one. Since fuel equals gravity times time, minimising fuel and minimising time are the same problem, so the answer is to fall as fast as you dare and brake as late as you dare. That is the suicide burn, and it beats a steady descent by about twenty-five to one.
Why is hovering so expensive?
Because it is pure time with no progress. Twenty seconds of hover costs exactly gravity times twenty, no matter where you do it or what happens afterwards. A pilot who slows down early to feel safe burns the fuel they needed for the landing itself.
What is the stopping distance for a lander?
Velocity squared divided by twice the net deceleration, where net means thrust minus gravity. Start braking at exactly that altitude and you arrive at the ground at zero speed — which the tool draws as a dashed line so you can see it coming.
What if the engine is weaker than gravity?
Then you can never stop, at any altitude. The stopping distance is infinite and no throttle pattern fixes it. A lander needs a thrust-to-weight ratio above one before any of this arithmetic means anything.
How fast do you hit the ground in free fall?
The square root of twice gravity times height. From 500 m on the Moon that is 40.2 m/s. The same drop on Earth arrives at 99 m/s, which is why the Moon is the forgiving place to land despite having no atmosphere to help you.
Does the identity account for the rocket getting lighter?
No — it assumes constant mass. A real rocket loses mass as it burns, which helps slightly and does not change the shape of the answer: time is still the thing you are paying for.
Does this send anything anywhere?
No. The simulation runs entirely in your browser.
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