Teaching model. A point-mass physics model for a humanitarian-style cargo airdrop, with textbook-level numbers. It is not a certified airdrop planning tool.
Terminal velocity. Weight balances drag on the load plus canopies:
vt = sqrt( 2 m g / (rho CdA) ), CdA = eta n Cd (pi D2/4) + CdApallet
Air density rho follows the International Standard Atmosphere for the drop-zone elevation plus the temperature offset, so a high, hot zone falls faster. eta < 1 is a simple cluster-interference loss.
Equation of motion. Integrated with RK4 and sub-stepping, using the velocity relative to the moving air:
m dv/dt = m g - 1/2 rho(z) CdA(t) |v - w(z)| (v - w(z))
Inflation and opening shock. After the static line stretches at speed vs each canopy fills over a fill time tf = nf D / vs (nf about 8 diameters of air flow for a flat circular canopy). Drag area grows as (tau/tf)2. The peak force Fmax occurs while the heavy load is still fast. The opening-shock factor is Cx = Fmax / (qs CdAfull) with qs = 1/2 rho vs2. Heavy loads slow down while filling, so Cx is below the infinite-mass value. Staggering a cluster or reefing (first stage at a fraction of the area, then disreefing) lowers the peak.
Wind profile. Power law u(z) = u10 (z/10)alpha or logarithmic u(z) = u10 ln(z/z0) / ln(10/z0), held constant above 500 m. Direction veers with height. The load drifts with the air it is in, so the landing offset is roughly the integral of the wind over the descent time.
Computed air release point (CARP). The aircraft flies a ground track with airspeed Va (ground speed Vg = wpar + sqrt(Va2 - wperp2)). The planner simulates the drop from the origin with the forecast wind, takes the landing offset D and releases at R = marker - D. A wrong forecast shifts the landing by about (wtrue - wforecast) x descent time.
Ground impact. A crush layer with constant crush force Fc = m ac stops the load over s = v2 / (2 (ac - g)). If s exceeds the available stroke the layer bottoms out and the residual speed is absorbed by a stiff structure, giving a large spike in g.
Simplifications. Point mass (no swing or oscillation), constant Cd, no gusts within a run, flat ground, pallet extraction idealised as a delay, horizontal landing motion ignored. Random runs vary wind speed (log-normal), wind direction, deployment delay and release timing.