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One of the more unintuitive facts about sailing is that it is possible to harness the power of the wind to sail in a direction against that of the wind or to sail with a speed faster than the wind itself, even when the water itself is calm.  It is somewhat less known, but nevertheless true, that one can (in principle) do both at the same time – sail against the wind (even directly against the wind!) at speeds faster than the wind.   This does not contradict any laws of physics, such as conservation of momentum or energy (basically because the reservoir of momentum and energy in the wind far outweighs the portion that will be transmitted to the sailboat), but it is certainly not obvious at first sight how it is to be done.

The key is to exploit all three dimensions of space when sailing.  The most obvious dimension to exploit is the windward/leeward dimension – the direction that the wind velocity $v_0$  is oriented in.  But if this is the only dimension one exploits, one can only sail up to the wind speed $|v_0|$ and no faster, and it is not possible to sail in the direction opposite to the wind.

Things get more interesting when one also exploits the crosswind dimension perpendicular to the wind velocity, in particular by tacking the sail.  If one does this, then (in principle) it becomes possible to travel up to double the speed $|v_0|$ of wind, as we shall see below.

However, one still cannot sail against to the wind purely by tacking the sail.  To do this, one needs to not just harness  the power of the wind, but also that of the water beneath the sailboat, thus exploiting (barely) the third available dimension.  By combining the use of a sail in the air with the use of sails in the water – better known as keels, rudders, and hydrofoils – one can now sail in certain directions against the wind, and at certain speeds.  In most sailboats, one relies primarily on the keel, which lets one sail against the wind but not directly opposite it.  But if one tacks the rudder or other hydrofoils as well as the sail, then in fact one can (in principle) sail in arbitrary directions (including those directly opposite to $v_0$), and in arbitrary speeds (even those much larger than $|v_0|$), although it is quite difficult to actually achieve this in practice.  It may seem odd that the water, which we are assuming to be calm (i.e. traveling at zero velocity) can be used to increase the range of available velocities and speeds for the sailboat, but we shall see shortly why this is the case.

If one makes several simplifying and idealised (and, admittedly, rather unrealistic in practice) assumptions in the underlying physics, then sailing can in fact be analysed by a simple two-dimensional geometric model which explains all of the above statements.  In this post, I would like to describe this mathematical model and how it gives the conclusions stated above.