A gliding mammal as a point mass with a membrane wing: lift, drag, a full lift-drag polar and the equilibrium glide, integrated with RK4.
The science
Forces. Treat the animal as a point mass m with a membrane (patagium) of area S. Lift and drag act perpendicular and parallel to the airspeed V: L = ½ρV²S CL, D = ½ρV²S CD. The drag polar is CD = CD0 + CL²/(πeAR) with Oswald factor e = 0.8 (membrane wings are less efficient than rigid ones; the value is an illustrative assumption).
Equations of motion. With flight-path angle γ (negative when descending) and horizontal wind w: m dV/dt = −D − mg sinγ, mV dγ/dt = L − mg cosγ, ground velocity (V cosγ + w, V sinγ). Integrated with RK4 at 5 ms.
Equilibrium glide. When accelerations vanish, tan(−γ) = D/L = CD/CL and V = √(2mg / (ρS CL cosγ)). The best glide ratio occurs at CL* = √(πeAR CD0), giving (L/D)max = ½√(πeAR / CD0). The posture-held CL slider lets you fly above or below it.
Numerical checks. (1) Started exactly in equilibrium (V and γ from the formulas) in still air, the RK4 trajectory must keep descending at the glide angle: range/height equals L/D. (2) Energy: the loss of potential plus kinetic energy equals the work done by drag, ∫ D V dt, from the stored trajectory. (3) With lift and drag set to zero the integrator must reproduce the ballistic range v0√(2h/g). A real jump is not in equilibrium: it begins with a dive and the speed oscillates (the phugoid), so the real range is shorter than (L/D)·h; the page shows this honestly.
Honest simplifications. Real colugos (about 1–1.5 kg, membrane roughly 0.2–0.3 m²) steer by changing membrane camber and tail posture, so CL and CD0 change during a glide; here CL is held constant, the animal is a point mass, the air is still apart from a constant horizontal wind, and the numbers (e, CD0, reported ranges up to 100 m or more) are illustrative values, not field measurements. The landing is not modelled.