Institute for Physical AI @ BMI · a demonstration
On land, twice the push roughly means twice the speed. In water it does not — and the reason is a rule every swimmer feels in their arms. Watch the robot cruise, then double its thrust and see how little it gains. The gold ring is the board it circles; the source runs the real drag law, so it cannot lie to you.
Provenance. A body moving through water feels a drag force that grows with the square of its speed: ½·ρ·C_d·A·v². At a steady swim, thrust equals drag, so v = √(thrust ÷ ½ρC_dA) — speed rises only with the square root of the push. Double the thrust and you gain a factor of √2 ≈ 1.41; to go twice as fast you must push four times as hard. The board integrates thrust against this drag live. All numbers computed by the Institute's open swim core. The 3D model is illustrative; the numbers are exact.