motion.ts

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))