Momentum Drift & Stopping Distance
Fly a ship with no brakes, plus the physics: stopping time is linear in speed and stopping distance is quadratic, which is why you always overshoot.
Arrow keys or WASD: left and right turn, up thrusts. There are no brakes — to slow down you turn round and burn. The pale ring shows how far you would travel before stopping from your current speed, turn included. Reach the green circle without hitting a rock. Nothing is uploaded.
Stopping time is linear in speed. Stopping distance is not.
With thrust a and no drag, a full stop takes v/a and covers v²/2a. One is linear in speed and the other is quadratic, so they come apart — and people judge braking by the time, because time is the part you can feel. At the thrust (120) and turn rate (3 rad/s) this game uses:
| Speed | Time to stop | Burn distance | Drift while turning | Total |
|---|---|---|---|---|
| 50 | 0.42 s | 10 | 52 | 63 |
| 100 | 0.83 s | 42 | 105 | 146 |
| 200 | 1.67 s | 167 | 209 | 376 |
| 400 | 3.33 s | 667 | 419 | 1086 |
| 800 | 6.67 s | 2667 | 838 | 3504 |
From 50 to 800 is 16 times the speed, 16 times the time and 256 times the distance. The number you are watching grows in step with your speed; the number that hits the rock grows as its square.
And you cannot start slowing until you have turned round
Rotating does not slow you at all. Every second spent swinging the nose about is a second at full speed with the engine pointing uselessly — dead distance before the burn even begins. Half a turn at 3 rad/s takes just over a second, which at speed 400 is 419 units against the 667 the burn needs.
But the share the turn takes runs the other way from the total: 83% of the stopping distance at speed 50, 24% at speed 800, because drift grows linearly while the burn grows as the square. The turn is what catches you out when you are crawling; the burn is what catches you out when you are flying.
A small exact surprise
Burn until your velocity is exactly reversed — from +v to −v — and you end up precisely where you started. It takes 2v/a, and the displacement over that time is v(2v/a) − ½a(2v/a)² = 0 exactly. You overshoot by v²/2a and come back by the same amount, so reversing costs you nothing in position and everything in time — which is the resource nobody budgets.
The formulas are derived, and then the manoeuvre is flown step by step and compared against them: at speeds 100, 200, 400 the simulated stopping distance is within 0.060% of the predicted one, and a live check at speed 313 — a speed the table does not list — comes in at 0.019%.
How to use
- Arrow keys or WASD — left and right turn, up thrusts.
- There are no brakes: to slow down, turn round and burn.
- The pale ring shows how far you need to stop, turn included.
- Reach the green circle without hitting a rock.
- Press Full stop to watch the manoeuvre flown properly.
Frequently asked questions
What is the stopping distance formula?
Under constant thrust a from speed v, a full stop takes v divided by a and covers v squared over 2a. The time is linear in speed and the distance is quadratic, which is the whole reason people misjudge it.
Why do players always overshoot?
Because they judge braking by how long it will take, and time is the part you can feel. Going from speed 50 to speed 800 is 16 times the speed, 16 times the time — and 256 times the distance. The number you are watching grows in step with your speed; the number that hits the rock grows as its square.
Does turning round cost anything?
A great deal, and it is easy to forget. Rotating does not slow you at all, so every second spent swinging the nose about is a second at full speed with the engine pointing uselessly. At speed 400 that is 419 units of dead distance before the burn even starts, against the 667 the burn itself needs.
Does the turn matter more at high speed?
As a share of the total it matters less, which catches people out both ways. Drift grows linearly with speed while the burn grows as the square, so the turn is 83 per cent of your stopping distance at speed 50 and 24 per cent at speed 800. The turn is what catches you when crawling; the burn is what catches you when flying.
What happens if I burn until my velocity reverses?
You end up exactly where you started. Reversing from plus v to minus v takes 2v over a, and the displacement over that time works out to precisely zero — you overshoot by v squared over 2a and come back by the same amount. Reversing costs nothing in position and everything in time.
Are the numbers on this page derived or simulated?
Both, deliberately. The formulas are derived, and then the manoeuvre is flown step by step and compared against them. At the speeds in the table the simulated stopping distance is within 0.06 per cent of the predicted one.
Does this send anything anywhere?
No. Every field is generated and every flight computed in your browser, and nothing is uploaded.
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