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.

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A similar question is, how is it even possible that an airplane can fly horizontally but will quickly stall out and hit the ground if it tries to fly straight up? Maybe I’m the only one who has the initial intuition that straight down ought to be the best direction to push if one wants to go up. I mean, I also kind of have the intuition that the wings help somehow when you go horizontally, but mathematically, how? It’s not like they’re flapping or something; they’re just sitting there, consuming no fuel.

Most people will answer something about the asymmetrical shape of the wings and Bernoulli mumble mumble, but that can’t be the whole story because, first of all, planes can fly upside down just fine, and second, it feels like you might hope for an answer at a more basic Newtonian level.

For me, the key observation was that in horizontal orientation the wings push a lot of air down a little bit, while in vertical orientation the propeller pushes a little bit of air down a lot. And in general, for a given energy expenditure, you achieve a greater change in momentum by accelerating a large external mass a little bit than by accelerating a small external mass a lot.

The mathematical reason for that is basically that momentum conserves m*v while energy conserves m*v^2. So if you’ve got some m*v^2 stored up in the form of combustible fuel or a flywheel or something, you can get more m*v out of it by using a small v and big m.

That’s also why, when you tread water, you have to move your hands horizontally like you’re spreading butter. If you just push your hands straight down you accomplish little. And it’s also sort of why it’s easier to stand on the ground than to fly — the Earth is a lot heavier than the volume of air you can reach.

Nice observation(s)! Although I’m not sure exactly how to fit your last point about standing being easier than flying into it: I definitely agree that this explains why jumping two feet is easier than raising yourself two feet by flapping your arms.

But I feel like this kind of analysis will never directly explain situations where staying still takes energy: indeed, standing on the ground takes no energy, maintaining some fixed height by flying takes some energy, but maintaining a position in space very far away from the earth again takes almost no energy. I feel like those determining those facts actually require examining the forces involved more closely, rather than just analyzing momentum and kinetic energy