I guess I did not understand your posting correctly. Anyway, let’s see if I can clarify. The air does not push directly and perpendicularly the surface of the sail as some may think. If you want to sail downwind and just put a barn door perpendicular to the direction of the wind, the effect would be dismal because it is stalled. For a sail to work the air has to flow along its surface on both sides, just like the wing of an airplane. The air is deflected creating a force against the sail.
Now… if I am sailing directly downwind, the fastest I could hope to go would be with the wind (I hope this is obvious)and this would be a theoretical, unreachable, limit.
Yet, If I am sailing across the wind, there is no reason why the 10 knot wind cannot give me a 15 knot speed across it. Think of a (triangular) narrow wedge of ice resting on its side on the kitchen counter against the wall. Now you push with a chopstick the ice against the wall and because of the wedge shape the ice will slide sideways. If the angle of the wedge is 45 then the ice will slide sideways the same distance you pushed the stick. If the angle is less than 45 then the ice will slide sideways more distance than you pushed the stick in. See? A sailboat is very similar.
Of course the conservation of energy and momentum still apply. The sail deflects a mass of air backwards (or sideways from the view of the wind) and generates a force in the opposite direction.
If you think of the above example but now point the chopstick at an angle you can see how a boat can sail at an angle of less than 90 to the wind and therefore sail upwind by tacking.
While sailing looks pretty complicated (and it is)the basic thing you are doing is trimming the sails to maintain a constant angle of attack no matter where the wind is coming from and other variables.
All this got me thinking:
The wing of an airplane, the rotor of a helicopter, the sail of a boat, the keel of a boat, the rudder of a boat, the blade of a propeller, the blade of a fan, the blade of a turbine… they all work under exactly the same principle: the fluid is displaced and the reaction creates a force on the solid object called “lift”. (Yes, even the force generated by the keel of a boat is called lift.) This force is always perpendicular to the surface and at an angle to the direction of travel of the fluid.
BTW, the keel of a boat is another good example of a symetrical object with equal lengths on both sides, which generates lift. The different length of travel cannot explain it either.
Any sailor can tell you a sail affects the air as it passes. A boat to windward will rob you of wind and you always want to be the one to windward.
The air speeds up as it goes by the sail but leaves the sail upwind of where it would be if there was no sail: It has been slowed down and deflected sideways (from the view from the wind).
Another thing many people have trouble understanding: When you are sailing, a freewheling propeller creates more drag than a locked one because the locked one is stalled while the turning one is creating drag.