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Model check
The science
The model: a point mass with a drag polar
The paper plane is treated as a point of mass m moving in the vertical plane with a fixed lift coefficient CL (set by how the wings are folded and trimmed). With the air-relative velocity va = v − w (wind w, thermal as vertical wz), the aerodynamic forces are
L = ½ ρ va2 S CL (perpendicular to va), D = ½ ρ va2 S CD (along −va), CD = CD0 + K CL2, K = 1/(π e AR)
and m dv/dt = L + D + m g, integrated with fourth-order Runge-Kutta. The quadratic drag polar is the standard first approximation: zero-lift drag from skin friction and the folds, plus lift-induced drag that grows as CL2/AR, which is why low-aspect-ratio darts glide steeply.
Steady glide, best glide and the phugoid
In an equilibrium glide, lift balances the component of weight across the path and drag balances the along-path component: tan γ = −CD/CL, so the glide ratio is L/D = CL/CD, independent of weight, and the speed is V = √(2 m g cosγ / (ρ S CL)). The best glide ratio occurs where the line from the origin is tangent to the polar, at CL* = √(CD0/K). A throw faster or slower than the equilibrium speed trades speed for height in a long oscillation, the phugoid, whose period for a clean glider is the Lanchester value T ≈ π √2 V/g; a hard throw upward at high CL can carry the plane over the top in a loop. The range graph shows how the distance depends on throw speed.
Wind and thermals
The plane responds to the air, not the ground: a headwind lowers the ground speed and shortens the range, a tailwind extends it, and rising air of speed wz larger than the sink rate V sin|γ| lets a plane climb or hold height (soaring).
Honest simplifications
(1) The attitude is not simulated: CL is a constant you choose, so there is no stall, no pitching oscillation and no instability; real paper planes need correct trim (centre of gravity and the elevator folds) to behave like this. (2) Drag and lift coefficients are constants although the Reynolds number of a paper plane (about 104) changes them. (3) The wind is uniform; there are no gusts or ground effect. (4) Throw geometry is a point release. The drawn plane is enlarged for visibility.
Model checks in this page
(1) launched at the equilibrium speed and angle, the simulated distance per metre of height lost equals CL/CD; (2) the simulated phugoid period of a clean glider matches Lanchester's π√2 V/g; (3) with no drag the total energy ½v2 + g z is conserved.