The amazing science of flight is largely governed by Newton's laws.
Consider a wing cross-section:
Air hits it at a certain speed. However, the shape of the wing forces air to rush over it and under it at different rates. The top curve creates a partial vacuum - a region "missing" a bit of air. So, the pressure (force/area) on top of the wing can become less than the pressure below. If the numbers are right, and the resulting force below the wing is greater than the weight of the plane, the plane can lift.
This is often embodied as the Bernoulli Principle:
Pressure in a moving stream of fluid (such as air) is less than the pressure of the surrounding fluid.
A related concept is the equation of continuity:
A1 v1 = A2 v2
Think about a garden hose with your thumb over the end - the water comes out faster, right? The product of the area (A) and speed (v) through a tube is constant - make the area smaller and the fluid comes out faster.
Yet another way to think of flight is to imagine the wing above, but slightly inclined upward (to exacerbate the effect). There is a downward deflection of air. The reaction force from the air below provides lift and the lift is proportional to the force on the wing.
In practice, it works out to be:
Lift = 0.3 p v^2 A
where p is the density of air, v (squared) is the speed of the plane, and A is the effective area.
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