Until this morning, I didn’t really understand how it was possible for a sailboat to sail into the wind: popular descriptions like Wikipedia’s talk about keels and lift and the Bernoulli effect and so forth, but this feels like a leap beyond my understanding: I wanted an account of how it is even possible in terms of the very basics.
I think to most physics people this is obvious, but in case there are others like me out there, I thought I’d record the explanation I came up with this morning here.
Suppose we model the situation as follows: there is a single air particle with mass and velocity . The sailboat is also a particle; it has mass and is initially at rest. The two particles interact in some way (it could involve rudders, keels, whatever), and afterwards the air particle has velocity and the sailboat has velocity .
If we require that momentum is conserved and kinetic energy doesn’t increase, so that and , then it is possible to prove that our everyday intuition is correct! That is, you can show that indicating that the cosine of the angle that the sailboat makes with respect to the initial air velocity must be nonnegative: that means the sailboat can’t go against the wind! (In fact, if the sailboat’s velocity is nonzero, the dot product has to be strictly positive.)
The resolution is to add a third particle representing the ocean (or other air particles). Now it becomes possible for the sailboat’s final velocity to go against the wind, even though it is still impossible for the combined system of the ocean plus the sailboat to go upwind.
Looking at it this way with three particles, it’s actually pretty easy to see that it’s possible for the sailboat to go against the wind: imagine a bowling ball representing the ocean floating in space, with a ping pong ball representing the sailboat touching it. Another ping pong ball representing the air collides with the first ping pong ball. Both ping pong balls bounce in the opposite direction (and the bowling ball is deflected very slightly): The sailboat is now moving in the direction opposite the initial air velocity!
Anyway, I still have no idea how the rudders and keels and so forth actually work, but looking at it this way satisfied my curiosity about how it was even possible for a sailboat to sail against the wind.