A flying disc is a spinning wing. Throw it through a six-degree-of-freedom model with lift, drag, pitching and rolling moments and see why spin keeps it flying and why discs turn over and fade.
The science
Aerodynamic force
With air velocity V relative to the disc and angle of attack alpha (angle between the flight direction and the disc plane, positive when the air hits the underside):
F = 1/2 rho V^2 A (C_L l - C_D v) + m g
C_L = C_L0 + C_La alpha C_D = C_D0 + C_Da (alpha - alpha0)^2
l is the unit vector along the disc normal made perpendicular to the flight direction, A = pi R^2. Beyond about 20 degrees the coefficients blend smoothly into a flat-plate model (C_L = 1.1 sin 2 alpha), which is what makes a stalled or tumbling disc fall.
Moments and Euler's equations
M_pitch = 1/2 rho V^2 A d (C_M0 + C_Ma alpha + C_Mq q d/2V)
Spin adds a roll moment about the flight direction (C_Rr) and damping (C_Rp, C_Nr). The body rotates according to
I w' + w x (I w) = M I = diag(I_d, I_a, I_d)
with the orientation as a quaternion. Everything is integrated with RK4 at 600 steps per second.
Why spin stabilises
A disc spinning with angular momentum L = I_a w_spin does not tip when a torque M acts; the axis precesses at
Omega = M / (I_a w_spin)
so doubling the spin halves the precession rate and the disc keeps its attitude. At very low spin the same torque flips it over. The readouts compare this formula with the axis rate measured from the simulated orientation.
Turn and fade
At the small angles of attack of a fast throw the pitching moment is nose-down (C_M0 < 0). For a clockwise-spinning disc the resulting precession banks the axis toward the right (anhyzer): the disc turns over to the right. Later, as speed drops and alpha changes, the nose-up term C_Ma alpha takes over and it fades back to the left. Understable discs have a more negative C_M0, overstable ones a less negative or positive one. A headwind raises the effective airspeed, so it makes any disc behave more understable.
Vacuum check
With no air the motion is a pure projectile; from height h: R = v cos(g) (v sin g + sqrt(v^2 sin^2 g + 2 g h)) / g. Experiment 1 compares the integrator with this formula.
Simplifications
Coefficients follow published frisbee-style sets (Hummel 2003, Potts and Crowther 2002) rounded for the ultimate disc; driver and putter values are plausible scalings tuned for realistic distances, not measurements of specific moulds. No ground effect, no wind gradient, no Magnus-type spin lift, constant coefficients in Reynolds number, and landing is a simple stop.