I need the most simple lift equation that once solved with Mathcad gives a realistic vertical speed of a plane and implicitly the altitude h(t).
Also the drag eq. (1) leads to a good solution, a Vh(t) that increases and finally reaches a limit (the maximum horizontal speed) and stays there during the duration of the flight, the lift eq. (2) stabilizes at a Lift(t) - m * g = ct. > 0 and in consequence Vh(t) keeps growing indefinitely because an eq. of the type m * dVv(t)/dt = ct. leads to a solution Vv(t) that rises linearly with time.
Question: It is quite clear that the vertical speed of a plane, Vh(t), can not grow indefinitely. How can I stabilize it to o constant value. What do I have to add in the lift eq.?
Drag eq.: m * dVh(t)/dt = T - Drag(t) (1),
Drag(t) = 0.5 * Cd * r * S * ( Vh(t) + Vw(t) )^2,
Lift eq.: m * dVv(t)/dt = Lift(t) - m * g (2),
Lift(t) = 0.5 * Cl * r * S * ( Vh(t) + Vw(t) )^2,
- Vh(t) = horizontal speed, Vv(t) = vertical speed, both of them have to be determined being unknown functions.
- known parameters: m = the mass of the plane, r = air density, S = the wing surface, T = thrust = ct., Cd, Cl are the drag and lift coefficients, g = 9.81 m/s^2, Vw(t) = the wind speed, that is usually a known constant but can have other forms given as functions of time, t.