When sailing "with" the wind, meaning traveling in the same direction as the wind, there are two main forces acting on the sailboat, a force on the sail by the wind and a frictional force on the hull of the sailboat. It is the relative balance between these two forces that determines the motion of the boat. These forces are illustrated in figure 1. If a wind has just arisen, the force from the wind on the sail will be greater than the frictional drag on the boat hull, and the boat will begin to move in the same direction as the wind. As the boat accelerates, the frictional drag increases until the forces are in equilibrium and the boat moves at a constant velocity. If the wind then begins to decrease, the frictional force will dominate causing the boat to slow down. This was the mechanism behind ancient sailing ships and had the major disadvantage of limiting oneself to traveling in the direction of the wind. Most likely, the sails were only hoisted in the case of an advantageous wind, in order to spare the weary oarsmen. Later, a rudder was added to sailing vessels which helped aid in steering, but it was still necessary to have a component of the wind in the direction one wanted to travel.
The case of sailing, when the wind is no longer "at your back" is when the physics gets more interesting. Actually, it is not possible to sail directly into the wind, usually only about an angle of 40-45o is possible. A few racing sailboats manage to cut this angle to as low as 35o , but this is an extraordinary case. When sailing at an angle into the wind, the sails function is no longer to "catch" the force of the wind. The sail actually acts in the same way as an airplane wing that has been tilted on its side. Figure 2 shows the streamlines around a sail, looking down from above. If we apply Bernoulli's theorem to the sail, referring to figure 2, we have the air passing along the right side of the sail must travel faster than the air traveling along the inside if the sail. This results in a pressure gradient across the sail, and a pressure gradient force (PGF) oriented perpendicular to the sail face. Figure 3 shows the forces on the sail. The pressure gradient force across the sail is indicated by the vector labeled "sail lift" in figure 3. There is also a frictional drag force, acting parallel to the sail, resulting from the wind passing over the sail. The resultant of the pgf and the drag force represent the actual force on the sail. This force can then be decomposed into a component in the direction of the boat's motion (a driving force), and a component perpendicular to the motion (a sideways force). The driving force is the force that propels the boat forward in the direction of motion. The sideways force will tend to push the boat off course, or even flip the boat over. This is where the keel comes into play.
From Kundu (page 575), "The keel...is a thin vertical surface extending downward from the bottom of the hull.” The keel will serve the purpose of compensating for the sideways force on the sail. Figure 4 shows the streamlines around the keel of a sailboat. looking up from the bottom of the keel. Notice that the keel is directed at a small angle off the actual track of the sailboats motion. This angle, called the yaw angle or angle of attack, is usually about 2-3o , and is essential for the keel's purpose. This angle creates a situation such that, for figure 4, water passing on the left side of the keel moves faster than water moving over the right side. From Bernoulli, a PGF develops giving a "sideways force" on the keel acting in the opposite direction to the sideways force on the sail. As you might imagine, at times, this sideways force may not always be sufficient to compensate for the sideways force on the sail. Therefore, gravity is also used to help compensate in the form of a lead ballast at the bottom of the keel, or, in smaller boats, the sailors using there own weight by hanging off the side of the boat.
How can we get the boat to move faster? If we put a smaller sail, called the jib, in front of the mainsail, we can effectively increase the speed of the wind passing over the "front" of the mainsail. Figure 5 shows this "slot effect." If we can increase the speed over the front of the sail, we can increase the pressure gradient force, and therefore increase the forward "lift" on the sail and make the boat move faster.
