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Height of the ball above the floor against time, with the flat-floor (uniform gravity) parabola dashed
Where the ball lands against launch angle: rotating station against the uniform-gravity prediction, hoop position in green
Apparent weight of a runner against running speed along the floor (spinward positive)
Drop test: backward shift of a released ball against release height, exact against the small-angle formula
Starting.
running...
The science
Artwork: the hub, spokes, hull panels and ball are Blender-rendered images drawn over the physics. They are decoration: the motion comes only from the equations below, and the ring is not drawn to the real proportions of any station.
Spin gravity. A habitat that rotates at angular speed ω presses everything standing on its inner rim against the floor. In the rotating frame this appears as a centrifugal acceleration ω2r pointing away from the axis, so the floor gravity at radius R is
g = ω2 R ⇒ rpm = (60 / 2π) √(g / R)
For 1 g (9.81 m/s2) a radius of 224 m needs 2.0 rpm, a 60 m ring needs 3.9 rpm. Gravity falls linearly toward the axis, g(r) = ω2r, to zero on the axis (the zero-g zone), and a person 1.8 m tall feels a head-to-foot drop of 1.8 / R of the floor value (0.8% at R = 224 m, 3% at R = 60 m).
What a thrown ball does. Once released, the ball feels no force: in space it moves in a straight line. If it leaves the point (R − h0, 0) with velocity u relative to the floor, its velocity in space is V = u + ω × r0, i.e. it keeps the rim speed ω(R − h0) sideways. It lands when |r0 + Vt| = R, a quadratic in t that the page solves exactly; the landing point on the turning floor then follows from the angle ψ = atan2(Y, X) − ωt. Seen from the station, the same motion obeys
r̈ = −2 ω × ṙ − ω × (ω × r) (Coriolis + centrifugal)
The lab uses the straight-line solution to draw the motion and an independent RK4 integration of these rotating-frame equations as a check. The Coriolis term is the reason a ball thrown spinward comes down sooner and shorter than flat-floor intuition predicts, an anti-spinward throw stays up longer, and a vertical throw lands behind you. The Jacobi constant ½|u|2 − ½ω2r2 is conserved in the rotating frame.
Runner and drop test. A runner moving at speed u along the floor has inertial speed ωR + u, so needs a centripetal acceleration (ω + u/R)2R: apparent weight = (1 + u / ωR)2 of the standing weight. Running spinward is heavier, anti-spinward lighter, and at u = −ωR you would be stationary in space and weightless. A ball let go from height h (no speed relative to the floor) moves along a tangent and lands behind its release point by R(ωt − atan ωt), t = √(R2 − r02) / (ωr0), which for small h is about (2h)3/2 / (3√R).
Honest simplifications. (1) Motion is in the plane of rotation; motion along the axis is unaffected by the spin and is left out. (2) No air drag, no wind and no ceiling: the interior is treated as a large open cylinder full of still air, as in the O’Neill cylinder concept. (3) The station spins at constant rate, perfectly rigid and balanced. (4) The comfort labels (about 2 rpm comfortable, up to 4 rpm with adaptation, above 6 rpm severe) are rules of thumb from rotating-room studies, not medical advice, and depend strongly on head movements and training. (5) In the 3D scene the habitat is drawn at a fixed size and distances are scaled to it; people, the hoop and the ball are enlarged so you can see them.
Checks run in the page. The exact flight time and landing point are compared with the rotating-frame RK4 integration, the Jacobi constant is checked for drift, a gentle toss in a large ring is compared with the flat-floor 2v/g, the dropped-ball shift with (2h)3/2 / (3√R), the runner-weight formula with exact cases, and the 1 g design rule with 2.00 rpm at 224 m.