Goal: carry the parcel from the blue START pad to the green pad by the house and land softly. Hold UP to take off, hold RIGHT to tilt and fly.
Green line: push up. Red line: weight. When they match, the drone hovers.
Thrust and weight. A spinning propeller pushes air down, and the air pushes the drone up. One propeller makes T = CT ρ n² D&sup4; (n = turns per second, D = diameter, ρ = air density 1.2 kg/m³, CT ≈ 0.11 here). The weight is W = m g. If the four thrusts add up to more than W the drone climbs, if less it sinks, if equal it hovers. Bigger propellers move more air, so they push harder at lower speed.
Power to hover (momentum theory). To push air down at speed vi, vi = √(T / 2ρA) where A is the disc area πD²/4. The ideal power is P = T vi = T3/2 / √(2ρA). Real props waste some (figure of merit 0.45 here) and motors waste some (efficiency 0.6), plus 1.5 W for electronics. Big discs need less power for the same weight, so they fly longer.
Battery. Energy E = capacity × voltage (a 600 mAh, 7.4 V pack holds 4.4 Wh = 16 kJ). We use 85% of it, so time = 0.85 E / P. The voltage drops from 8.4 V to 6.6 V as it empties, so the motors get weaker and the helper must spin them faster. At 0% the toy cuts the motors and falls.
Tilting to move. Spinning the two rear propellers a little faster than the front pair tilts the drone forward. The thrust now points a bit forward, so a = (T/m) sinθ. To keep its height the drone needs T = m g / cosθ, which gives a = g tanθ. Air drag F = ½ ρ CdA v² limits the top speed.
Why two spin each way. Every propeller twists the drone the opposite way it spins (torque Q = P/2πn). Two clockwise and two counter-clockwise cancel out. Press "All the same way" to see the toy spin round.
Honest simplifications. The numbers (CT, efficiencies, battery curve) are typical toy values, not one real product. We ignore ground effect, wind gusts, propeller flapping and motor heating, and fly in one vertical slice only (left/right and up/down). The drone is drawn 6 times bigger so you can see it. Wind only pushes through air drag. Quadcopter toys are not connected with any brand.
Check. The page flies hidden test drones with the same code and compares: (1) hover time until the battery cuts out versus 0.85E / P from momentum theory, and (2) a climb with constant thrust versus ½(T/m − g)t².