It is known that the velocity of an unpowered yacht can be faster than the velocity of the wind driving the sails. Can anyone explain this phenomenon? I'm looking for the clearest and simplest explanation.
In zero resistance ideal conditions, is there a threshold velocity that the wind must exceed for this to work? It seems if the boat is faster than the wind, then the 0.5mv^2 of the boat, must be balanced by kinetic energy delivered by the air with a larger effective m than the boat. I can't see how this would be possible for low wind velocities. Thus I'm not sure your statement that tan theta can be arbitrarily small is the full picture.
For a detailed answer refer to: "Sailing faster than the wind"
http://en.wikipedia.org/wiki/Sailing_faster_than_the_wind
The yacht can even go in the direction opposite to the wind! Would you call it going faster than the wind driving it? (the drive is in the opposite direction)
Yes, you can in principle design a bicycle that goes forward when you pedal backwards. Also notice that the wheels of a bicycle can go faster than the pedals driving it. Can we parametrize the properties of yacht sails in such a way that we can express the phenomenon in terms of gear ratios? It seems to me we need a general way to describe all things that move faster than than their driving forces, in terms of gearing ratios and mechanical advantage.
The answer is very simple. The wind is a vector field and does not have one direction and one speed. A yacht's thrust is due to integration of the field's dot product with the sails' manifold. Hence, depending on orientation it is natural to expect the integration to be larger than one component of the wind's velocity vector. That's how sails works on the open seas.
I agree my language was imprecise, but I think the pedal bicycle analogy conveys my meaning.
Suppose you had a ball free to roll on a track, and it was being pushed by a moving wall. If the wall is directly behind the ball, and moving in the direction of the track, then the ball's speed will equal the wall's speed. Let's call this the zero degree angle, since the movements are perfectly aligned.
But suppose the wall is moving at an angle to the track. The ball now has to roll faster than the wall moves: at 45 degrees, sqrt(2) times faster. In general, the ratio is 1 over the cosine of the angle; as the angle approaches 90 degrees, this goes to infinity.
How does this relate to a boat? Think of the wind as the wall. Think of the boat's keel as the track, that forces it to go in a different direction from the wind. If we assume no friction, the boat could go extremely quickly when traveling at right angles to the wind.
The above analysis is overly simplistic. In particular, it does not account for the setting of the sail, which is important. But it does demonstrate how a keeled boat could travel faster than the wind pushing it. This is, in fact, the main reason that boats have keels.
@Howard. I like the simplicity of your answer. I was looking for the most elegant one. Your answer reminds me of the idea that the point of intersection of a pair of scissors moves faster than the handles.
My own favorite way of demonstrating this effect is to place a plastic ruler and a plastic right-angled triangle flat on a table. I put the hypotenuse of the triangle against the ruler. I hold the ruler still with one hand and push the short side of the triangle with my finger. The direction of my finger is perpendicular to the ruler.
The triangle moves much faster than my finger.
Derek, this is a great idea - the intersection point of two almost parallel lines moving in an opposite direction and remaining parallel to themselves (translation), moves in a perpendicular direction with many times higher velocity. As I recall from my school years (60's), the Russian physicist Perelman argued that this point can move at the speed of light. Then I discovered it alone when waiting in line at a post office, I watched the gap formed by the moving door. Later, as a graduate student, I tried to obtain patents for various scanning devices based on this idea
When multiplied, this effect is known as a Moiré pattern that was widely used in various mechanical advertising devices, transducers... This idea can be observed even in electronics when illustrating graphically the idea of the dynamic load in amplifying transistor stages.
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