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Knowing the angle of the sail and the angle of the wind, how can you calculate the resulting angle of the boats movement?

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I don't have time to make this a full answer, but break the velocity down into its components (do you know the velocity)? If not, break it into components using the velocity as a variable. Then, I am not sure how much you are supposed to read into the sail direction - what assumptions are you making? – analysisj Jan 1 '12 at 16:46
If the sail of the boat is at a certain angle, surely the boat will go at a different angle? It's like light refractoriness. – Derek Jan 1 '12 at 16:51
It's effect should vary based on the shape of the sail... – analysisj Jan 1 '12 at 16:54
It depends whether there is resistance against lateral movement - a keel or centreboard. Also whether the steering mechanism (rudder) can be ignored. And many other factors - e.g. sails are not flat. – Mark Bennet Jan 1 '12 at 16:56
Actually, for a real sailboat, the (first-order) answer is that it moves through the water in the direction of its keel. The relation between wind and sail determines whether it is speeding up or slowing down. For a second order approximation, the wind can also push the keel sideways through the water, but supposedly that effect is deliberately minimized by the hull design -- it needs to be minimal or the boat wouldn't be able to tack. – Henning Makholm Jan 1 '12 at 17:11

In a simple model (of a boat in stationary, frictionless water), there are three components to the motion of a sailboat:

  • The direction of its keel, $k$
  • The direction of the wind, $w$
  • The normal direction of its sail, $s$

Speaking without precision about physical quantities, the force on the sail is the component of the wind normal to the sail. The force on the boat is the component of the force on the sail parallel to the direction of the keel. So the force on the boat is $\langle w,s\rangle \langle s, k\rangle k$.

In a more complicated model, one must take into account any currents pushing the boat and the interaction between the boat's hull and the water.

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Nitpick: what you have there is not the "force on the boat", but the force after being partially canceled by sidewards forces on the keel. To this one must add/subtract friction on the hull as it moves through the water. And of course everything needs to be done in a frame of reference that moves with the current. – Henning Makholm Jan 2 '12 at 2:03
Yes, true. I have added a caveat to my answer. – Neal Jan 2 '12 at 18:01

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