Hot Air Balloon Lift and Control (2D): Altitude Dynamics and Wind Steering

Presets

Balloon and weather

Autopilot target

Live readouts

What to try

1. Hold Burn until the vertical speed turns positive; release and watch it drift back down as the envelope cools. 2. Switch the autopilot on and move the target altitude: it pulses the burner and vents to hold it. 3. Make the load heavier or the day hotter: the temperature needed to float rises. 4. Look at the plan view: at different altitudes the wind blows in different directions, so climbing or descending steers the balloon.

Independent numerical checks

The science

Buoyancy. The envelope is open at the bottom, so the hot air stays at ambient pressure: ρ = p/(R T). Net lift is L = (ρ_a - ρ_i) V g - m g, with m the structure, basket, fuel and passengers. Floating needs ρ_i = ρ_a - m/V, a closed-form temperature that the page checks.

Atmosphere. International Standard Atmosphere troposphere: T = T₀ - 6.5 z/1000, p = p₀(T/T₀)^5.256. Thinner air gives less lift per kelvin, so climbing needs ever hotter air.

Dynamics. (m + ρ_iV + ½ρ_aV) dv/dt = L - ½ρ_a C_d A v|v| (the last mass term is the added mass of the air the envelope pushes). Heat: c_p ρ_i V dT_i/dt = P_burner - UA (T_i - T_a) minus mixing when venting. The time lag of the temperature is why balloons answer a burn slowly.

Wind. Speed grows with height and the direction veers; the balloon moves with the local wind at its altitude and has no other horizontal propulsion. Simplifications: uniform envelope temperature, no envelope deflation or cooling gradient, constant burner efficiency, no rain, no thermals, terrain at sea level, standard-atmosphere pressure at the ground. Not a flight-training tool.