Lift, drag and the drag polar
In steady level flight lift equals weight, L = W = m g, so the lift coefficient is fixed by the airspeed: CL = 2W / (ρ V² S). Drag is modelled by the parabolic polar:
C_D = C_D0 + k C_L^2 k = 1 / (pi e AR)
D = 1/2 rho V^2 S C_D
P_el = D V / eta + P_hotel (eta = motor x controller x propeller)
Written in terms of speed, Pel = a V³ + b / V + Ph with a = ρ S CD0 / 2η and b = 2 k W² / (ρ S η). The cubic term is parasite (shape and skin-friction) drag, the 1/V term is induced drag from making lift. Their sum has a minimum.
Best endurance and best range
Endurance is battery energy divided by power, so it is longest at minimum power: dP/dV = 0 gives Vbe = (b / 3a)1/4, CL = sqrt(3 CD0 / k). Range per unit energy is V / P, longest at minimum P/V: Vbr = (b / a)1/4, CL = sqrt(CD0 / k), the maximum lift-to-drag ratio (L/D)max = 1 / (2 sqrt(k CD0)). A constant hotel load (camera, autopilot, radio) does not move the best-endurance speed but pushes best range higher: the root of 2 a V&sup4; − Ph V − 2b = 0. Small aircraft often have a best-endurance speed below the stall limit, so in practice you fly at about 1.2 Vstall.
Battery energy
Usable energy is E = fbatt m epack (1 − reserve), with pack-level specific energy epack in Wh/kg (a cell is about 250 Wh/kg, a packed and wired battery about 150 to 200). Endurance t = E / P and still-air range R = E V / P. Climbing adds W vz / η. Voltage sag and temperature are not modelled.
Wind triangle
The air carries the aircraft: ground velocity is air velocity plus wind. To hold a ground track the aircraft points its nose into the wind by the crab angle. With wind component along the track w∥ (positive = tail wind) and across it w⊥, the ground speed is Vg = w∥ + sqrt(V² − w⊥²) and sin(crab) = −w⊥ / V. Range over the ground is energy per distance times ground speed, so in a head wind w the best-range speed solves −2aV&sup5; + 3awV&sup4; + PhV² + 2bV − wb = 0: fly faster than in still air, because each second in the head wind covers less ground. The experiment tab checks this against the simulation. A lawn-mower pattern spends half its legs in a head wind and half in a tail wind, which never cancels exactly, because a slow ground speed costs more time than a fast one saves.
Camera: ground sample distance
For a pinhole camera looking straight down from height H with pixel pitch p and focal length f, one pixel covers GSD = H p / f on the ground. The footprint is (Nw GSD) × (Nh GSD). Photogrammetry needs the frames to overlap: line spacing is Fw (1 − side overlap) and the photo spacing along the line Fh (1 − front overlap); typical mapping uses 75 to 85 % front and 65 to 75 % side overlap so every ground point appears in at least three images. Motion blur in pixels is Vg texp / GSD; keep it under about half a pixel. A fixed camera mount crabs with the aircraft, so the footprint rotates and the effective side overlap shrinks in cross-winds; a yaw-stabilised mount removes that.
Survey path and turns
Lines are spaced evenly so the first and last footprints touch the boundary. Between lines the aircraft flies a turn at bank angle φ, radius R = V² / (g tan φ); if the spacing is less than 2R a three-arc bulb turn is flown. In a turn the load factor is n = 1 / cos φ, induced drag grows as n², and the stall speed rises by sqrt(n), which the energy integrator includes.
Radio link
Free-space path loss is FSPL = 20 log10(4 π d f / c) dB. The received power is Prx = Pt + Gt + Gr − L − FSPL; the link works while it stays above the receiver sensitivity plus a fade margin. Line of sight to a low antenna is also limited by the radio horizon, 4.12 (sqrt(h1) + sqrt(h2)) km with heights in metres.
What is simplified
Quasi-steady level flight with a constant polar (no Reynolds-number or flap effects), constant overall propulsive efficiency, air density from the ISA at mission altitude, no battery sag or temperature, perfect ground-track guidance with wind modelled as a mean plus a few sine gusts, turns flown on the planned ground track, no terrain relief in the energy model, and flat-earth free-space radio with no fading. The UAV, rail and ground station are drawn at real size; the ground control targets are enlarged for visibility. A generic, unbranded aircraft for education: this is not flight-planning software, and real operations must follow local aviation rules on height, line of sight and permissions.