motion.ts

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`a08492ba` 2026-09-07 Pavel Safronov	1	/**
	2	* How things fall in Tower's rotating frame. See design.md §1.9.2.
	3	*
	4	* Coriolis never pushes surface water sideways (§1.9.1), so the spin is invisible
	5	* in every static thing. The hull is sealed, so there are no stars to wheel
	6	* overhead either. That leaves what falls as the only evidence in the sky that
	7	* Tower turns at all, which makes these two angles load-bearing for the interior
	8	* view rather than decoration.
	9	*
	10	* Constant-g plus Coriolis is the standard approximation and is what these use.
	11	* It is exact only for drops small against the radius: gravity is 2% weaker 1 km
	12	* up, and the true inertial-frame path is a straight line, not a parabola.
	13	*/
	14	import { G0, OMEGA_RAD_S } from './constants.js'
	15
	16	/** Terminal velocity of an ordinary raindrop, m/s. */
	17	export const RAINDROP_TERMINAL_M_S = 9
	18
	19	/**
	20	* Lean of something already falling at a steady speed, in radians, antispinward.
	21	*
	22	* Coriolis gives 2ωv antispinward against g outward, and at terminal velocity drag
	23	* holds both constant — so rain leans by a fixed angle rather than curving. That
	24	* angle is small: 1.5° for an ordinary raindrop. Rain is a poor compass here.
	25	*/
	26	export const steadyFallLeanRad = (speedMS: number): number =>
	27	Math.atan((2 * OMEGA_RAD_S * speedMS) / G0)
	28
	29	/** Time for something dropped from `heightM` to reach the floor. */
	30	export const freeFallTimeS = (heightM: number): number => Math.sqrt((2 * heightM) / G0)
	31
	32	/**
	33	* Antispinward drift of something in free fall from `heightM`. Integrating the
	34	* Coriolis acceleration 2ω(gt) twice gives ωgt³/3.
	35	*/
	36	export const freeFallDriftM = (heightM: number): number =>
	37	(OMEGA_RAD_S * G0 * freeFallTimeS(heightM) ** 3) / 3
	38
	39	/**
	40	* Lean of the path of a free fall from `heightM` — where it lands, against how
	41	* far it dropped. This is the tilt of a waterfall's plume seen whole, and it is
	42	* the column design.md §1.9.2 tabulates: 7.6° over a kilometre.
	43	*/
	44	export const freeFallPathLeanRad = (heightM: number): number =>
	45	Math.atan(freeFallDriftM(heightM) / heightM)
	46
	47	/**
	48	* Lean of the velocity at the bottom of the same fall — the direction a streak
	49	* of it is moving as it arrives, and so the angle rain and spray are drawn at.
	50	*
	51	* It reduces to atan(ωt), independent of g, and it is exactly half again the path
	52	* lean in tangent: the drift integrates to ωgt³/3 over a drop of gt²/2, giving
	53	* (2/3)ωt, while the sideways speed reaches ωgt² against a fall speed of gt.
	54	* Both numbers are real and they describe different things, which is why
	55	* §1.9.2 says 7.6° for a kilometre drop and the interior view's readout says 11.3°.
	56	*
	57	* Unlike rain this keeps growing with the drop, so a tall waterfall is a compass
	58	* where a shower is not.
	59	*/
	60	export const freeFallLeanRad = (heightM: number): number =>
	61	Math.atan(OMEGA_RAD_S * freeFallTimeS(heightM))