Shot profile, rim-opening geometry for the entry angle, window-vs-angle curve and make probability vs launch angle.
Compares the engine speed with the vacuum closed form at four launch angles, and the measured clean window with 2(Rc-(r+rho)/sin a) at steep entry angles.
Not run yet.
The science
Flight. The ball (r = 0.1195 m, 0.62 kg) obeys m dv/dt = m g − ½ρA Cd|v|v + Magnus with CL = 1/(2+1/S), S = rω/|v|, integrated with RK4 by the same module as the 3D lab. The launch speed is solved so that the ball centre passes through the hoop centre; the page shows the side profile through the shooter–hoop line.
Vacuum check. Without air a ball released from height h0 at angle θ reaches horizontal distance s and height difference Δy with speed v = √( g s² / (2 cos²θ (s tanθ − Δy)) ); the minimum-speed angle is θ = atan((Δy + √(Δy² + s²))/s) (about 51.4° for a free throw from 2.1 m). The experiment compares the engine against these formulas.
Rim window. The rim has centreline radius Rc = 0.2366 m and tube radius ρ = 0.008 m. A ball entering at angle α below horizontal has a clear opening (along its line of flight) of w(α) = 2 (Rc − (r + ρ)/sinα), which closes completely at αmin = asin((r+ρ)/Rc) ≈ 32.6° and widens as the shot gets steeper. The inset draws exactly this geometry: the two tangent positions are the window edges. Steeper arcs give a bigger target but need more speed and are harder to control, so the best launch angle under release scatter lies a little above the minimum-speed angle.
Honest simplifications. Planar side view of a single shot (the 3D lab also covers off-axis shots and bank shots); fixed drag and restitution constants, simplified spin-coupled rim contact, no net physics beyond damping, a fixed release scatter model (1σ speed 1.2 %, angle 0.6°) for the probability scan; measured windows deviate from the closed form by a centimetre or so because rim bounces and spin are not in the formula.