Talk:High-performance sailing/Archive 3

1447:

Second, once the apparent wind direction goes forward of abeam, the only "wind" the boat has available from which to extract energy for forward motion is the vector component of the apparent wind which is perpendicular to the boat's direction of motion (the component parallel to the boat's direction of motion is from ahead and can therefore only produce drag unless the boat has a wind turbine driving a sub-surface propeller - but that is another article). Note, however, that since the induced wind is always parallel to the direction of the boats motion, that perpendicular component is the same as that available from the true wind, and therefore that component never changes. The only thing that changes is the direction and strength of the apparent wind, which in turn affects how efficiently the sails operate: it is easier to direct aft a wind coming from abeam than one coming from in front of abeam. The increasing strength (magnitude) of the apparent wind does not make available more energy from the wind, neither is it necessary, so there is no perpetual motion claim made here. The only issue is the perpendicular component of the wind from which energy is extracted, and the only limit to the speed of the boat is how efficiently it can convert that energy into a forward motion.

1165:

give the apparent wind vector. Your statement confuses induced wind and apparent wind. The induced wind (vector) is the opposite (negative) of the boat's motion (vector) over the water, and that induced wind vector is to be added to the true wind (vector) over the water to give the apparent wind relative to the boat. Since in this case they have opposite direction, adding true and induced winds as *vectors* is equivalent to the difference (i.e. subtraction) of their *magnitudes*. In an arbitrary reference frame, there are three velocity vectors: True Wind=Vtrue; Water=Vwater; Boat=Vboat. An observer (not the boat in question) floating in, and moving with, the water sees the true wind as . The boat in question moving downwind is moving as . The induced wind of the boat is the opposite (negative) of the boat i.e. . Adding (not subtracting) the true wind (1) and the induced wind (3) vectors: + = yields the apparent wind i.e. the wind that the boat experiences i.e. the wind that determines where the telltales on the boat point (direction) and what speed (magnitude) the anemometer on the boat measures. Also, if the boat is going straight downwind, there can never be a non-zero "remaining wind vector is perpendicular to the wind direction."

4371:

held full and pressurised by the existence of the rest of the sail creating a nice trailing edge for, and maintaining laminar flow over, this driving front part near the luff. This is what allows sailboats to do the apparently magical thing of making progress upwind. Overtaking the wind downwind is no more special than this. You are right that the key to it is to go so fast on a broad reach that you end up close hauled, but once you do, I'm sure, looking at that luff and imagining all the force-vectors sticking out of the forward-facing side of the sailcloth, it will make as much sense as when imagining them on a beat to windward. You can't overtake the wind in the same direction as the wind, but you can at an angle to it, provided you can sail very efficiently very close-hauled. So, my point is, there is no problem with the physics, we just need reliable sources that say people have done it, and hopefully to explain the physics as well (since we must not reference our own explanations). What I'm hoping to do is to convince Paul B that the physics is not screwy, so that he starts to help us find these references, rather than saying they can't exist. --

4662:

great deal of misplaced certainty but also in an ungracious manner. You have acted not so much, as you perhaps imagined, as the misunderstood prophet truthfully declaring the Holiness of God to a lost and sinful people, but more as the drunk in the middle of the street directing traffic. I note that you are a consultant and therefore expect you may wish to be careful about your web-public image; what you do next determines how you will be remembered for the future. If you apologize, all is forgiven (it will stroke our egos so much, that we actually taught someone else something, that we will forget what you said and how you said it), and welcome to the fold of those who have misapplied Sir Newton but realized their error (we're having jackets made up and there's a twelve-step program;-). If instead you are acting the part of a troll, then know that the game is up (see below), but looking at all the time others have spent responding to your posts you should consider it a job well done and know that you have perhaps helped improved their explanations on this topic. Either way, best regards. Your servant,

770:

adapted to demonstrate that this is possible on land. It's just inefficiency of props and turbines that make it hard to get something to work on water, as there's no fundamental law to say it can't work. In a thought experiment, this can be extended to make something work on water by say dropping and raising large sea anchors, and using energy from the wind through a gearing mechanism to winch the boat forward against the sea anchor, but necessitating the use of a complex device to drop sea anchors "upwind" and raise them with minimal energy loss if sustained progress is to be made. There are other possible variations of the same idea (really not so different from a turbine-prop in principle). Using a sea anchor to hold a boat in position with a turbine harnessing wind energy, storing the energy (ie in a battery) and then using it to propel the craft forward so that progress is made is fine, but this isn't continuous uninterrupted progress, and is somewhat unremarkable.

793:

and demonstrates that it is possible to advance directly into the wind while using only energy captured from the wind. What seems to puzzle people is the claim that it is possible to progress downwind faster than the wind. But the trick is to think in terms of apparent wind. High-performance boats go so fast, with respect to the true wind, that they are sailing upwind with respect to the apparent wind, even though they are going downwind with respect to the true wind. Seems paradoxical, but think of a boat that is on a 90-degree reach and has accelerated to the point where it is close-hauled with respect to the apparent wind. If that boat bears off just a little, would its speed start to drop off until it equalled the speed of the wind? No: why should there be a sharp discontinuity in the speed/course curve? In reality, the boat will accelate even more (unless it is already at hull speed), until it is again close-hauled with respect to the apparent wind.--

3645:

pushes the plane up, it flies because of a pressure difference between the top and bottom of the wing caused by Bernoulli's principle. For sailing craft that travel faster than the wind, the force generated on the airfoil by the apparent wind is an important part of the force driving the craft. It isn't a perpetual motion machine, first, because it won't work without wind, and second, it does have a limiting case where drag prevents further acceleration. I have been sailing for close to 50 years, and when I first heard that modern mult-hulls sail downwind with apparent wind forward, I didn't believe it. But I did some research, and read up on vector analysis, and saw the truth. Everyone has a bit of difficulty understanding how this can be, but it is. Did you read the Physics Today article I linked to, above?--

2527:

that a force applied over time is the same as momentum and may be an alternative trade to maintain conservation of momentum. The same goes for energy, which is a scalar (or magnitude only, non-vector) quantity. This is why the boat-fixed reference frame is often the easiest in which to make vector diagrams for analyzing sailing performance: the energy of the apparent wind in the boat-fixed frame does not change because the boat and sail are static and passive and have no external energy sources or sinks with which to give or take energy from the wind or water (ignoring frictional losses for the moment). The advantage that comes from this is simple: the magnitude of the momentum of the wind and the water in this frame do not change. Here is why:

4243:"In my equivalent frame, there is no apparent wind, only true wind". That is your fundamental fallacy. The sails see the apparent wind, not the true wind. If that were not the case, a boat sailing dead downwind with square sails would keep accelerating, becuase the true wind does not change. But you know perfectly well that a boat with square sails sailing dead downwind cannot sail faster than the wind: and this is precisely because the apparent wind drops to zero, so there is no more any force that can accelerate the boat. So the ONLY wind that matters is the apparent wind. Again, this is not OR, it is basic physics of sailing, explained in all textbooks, including those referenced in the material that you deleted.-- 4756:. If you do the analysis from the shore then the wind is the true wind. We all are assuming the true wind remains constant during our stopwatch-speed and balloon tests. In the shore-frame-of-ref the wind is always the same. In the boat it varies for every point of sail and every boat speed. The boat-frame-ref is difficult to analyse. The shore-frame-ref is EASY. But they are equivalent (=equally valid i.e. give the same results). That's basic physics. I choose the easy frame. If you choose the other frame I bet you cannot do the maths - no one here has done it so far. I can do the maths in the easy frame and I think you can follow. If you believe the assertion of the equivalence of the frames of ref. 4617:: Mr. Beardsell, it is your application of physics that is not up to the task. I am not insulting your physics literacy: I believe you understand the basic principles well enough to overcome your incorrect first impression of the physics involved. However, you do need to be more careful when you analyze the physics and not be limited by your first intuitive response (as so many of us have). My response was the same as yours and I had put out one foolish statement (elsewhere in one of the DDWFTTW threads) before I drew the simple diagram that convinced me. I have shown (above) in your preferred water-fixed reference frame - and can show in any reference frame - that the hull of a boat can be making VMG 1501:. If the wind had been 10 knots all the time (which was not the case) and if the downwind VMG had been equal to the wind speed, then USA 17 would have needed 2 hours to complete the downwind leg. That means that it would have completed the 20 mile upwind leg in 32 minutes, that is, at about 40 knots made good, meaning well over 50 knots over the ground. Of course USA 17 did not achieve such speeds upwind: anybody can watch the video and see the displayed upwind speeds, which were just over 20 knots. In fact, it took USA 17 63 minutes to complete the 20 mile downwind leg. Even if winds had been 10 knots (and they were less), the downwing VMG was nearly 2 times wind speed.-- 1650:

wind. The text and charts below show the force on the sails and the forward component of that force. It is not possible to determine theoretically whether the forward component at V2 would be sufficient to accelerate the boat further; but measurements of actual iceboat performances show that indeed it is. Iceboats accelerate until they are sailing at about 10 degrees (or less) off the apparent wind, and thus achieve speeds of 5 or more times the speed of the wind. This is the case even if the boat is sailing at 135 degrees off the true wind. So the VMG (progress in the dead downwind direction) is far greater than the speed of the wind.

2933:

direction. The other force is at right angles to that, the one that tends to tip the vessel to the side. When the set of the sail is such that the axial force is forward that drives the boat forward and vv. If on an upwind tack the keel/whatever is at 40 degs to the wind and the sail is at 20 degs then a proportion of the force experienced by the sail will be in a vessel-forward direction. If the keel is at 40 degs to the wind but the sail at 60 then there will be a backwards force. None of this contradicts the classical mechanics. We have found the component of the force necessary to explain the upwind sailing.

1360:

I'm not convinced that the forward force component cannot increase. You are correct that this would imply more power, but why can't the sail capture more energy from the wind at a tighter angle than it did at a broader angle? That would not violate conservation of energy and would not constitute perpetual motion. It seems to me that the force on the sail is proportional to the square of the speed of the apparent wind, so it could well be that the forward force component increases as the boat accelerates. As stated before, I will work out some numbers and post them at the bottom of the page, for further discussion.--

2523:

direction, the sail will experience a force in the opposite direction (Newton's first law plus the concept that a change in momentum per unit time is the same as a force). So, if instead of keeping the sail perpendicular to the true wind it is angled such that the back of the sail is further upwind, it is possible for the boat to be making VMGDW greater than the true windspeed and simultaneously to have the sail's intersection with a true wind streamline moving downwind more slowly than the true wind which means the sails are pushing against i.e. upwind the true wind and receiving a push downwind (VMGdwfttw_2.png).

2884:

the angle of the sail is such that the sail would push North on the calm air which means the air would push South on the sail. Anyone who claims to understand the equivalence of different reference frames but thinks this diagram, and the calm-air equivalent description in this paragraph, are not physically realistic needs to show the error(s) in the diagram above and provide a corrected diagram. Continuing with baseless claims that it is not possible with neither correct nor valid supporting arguments may still provide humor for many but is otherwise getting a bit old.

888:

that it is impossible to sail dead downwind faster than the wind, but I have not yet seen anybody explain why this is theoretically impossible, nor have I seen anybody explain why the thought-experiments that I provided are not correct. And please note that what I wrote is not original research, it is merely a summary of the various sources that I cited. In my view, it would have been more appropriate to flag the section and to discuss the matter further before deleting it. Regarding clean-up, I am all in favor of improvement. What specific items would you suggest?--

585:

sources - do you have others? Ref 31 repeats one of the refs I cite above - still only 1 possible verification. I disagree with your interpretation of ref 41 and point to the same author illustrating actual data in the second citation I make above. Another sentince in this section - "Of course real boats cannot equal those performances, although iceboats can come close to them." indicates the section is speculation, not reality. If tables are acceptable, they should be based upon real data, not hypothetical calculations based on a poor understanding of aerodynamics.

4714:

that I don't understand physics 101, especially when the physics in question can be found in any basic book on sailing. During the week end, I will try to produce the force diagrams that were mentioned earlier, and that show how an iceboat sailing at 135 degrees off the true wind will easily accelerate to a speed that results in VMG far greater than the wind. Those force diagrams will also show why there is no issue of conservation of energy. If that still does not convince Paul Beardsell, then we might have to envisage another approach.--

1121:

is created by the motion of the boat, but the motion of the boat is created by the TRUE wind, thus there is no perpetual motion. Think of a motor boat: it uses energy from its engine to create an apparent wind. That apparent wind could be used to drive a propeller-driven power generator. Similarly, a sailboat uses energy from the wind to create an apparent wind, which allows it to obtain even more energy from the wind. The point is that the energy required to drive the boat downwind always comes from the true wind.--

4538:

water after the sails and the keel have done their thing. Racing sailors are very familiar with 'dirty air' downwind of another boat, but to me that is all too complicated to visualise in such detail. In answer to your rhetorical question, the yacht is going faster than the south-going air, but in a south-westerly direction. By re-using the same magic that allows it to sail upwind, say 30 deg off the wind, it has now overtaken the southerly vector of the wind and is using it's 'upwind' ability to sail 30 deg off an

4068:

principles) allows us to analyse the system from any frame of reference guaranteeing the same outcome. That is why I suggest that the perspective of those in the boat be disregarded. We will look at this from the shore. The maths and the arguments become easier. In particular we can disregard APPARENT WIND as the only wind we will witness is TRUE WIND, i.e. wind relative to the earth. Those who do not follow this argument are disqualified from further comment.

1188:

shown from the various charts in the article (all derived from textbooks), there will be an apparent wind that can be forward of the beam. You cannot simply subtract the magnitude of the boat's speed from the magnitude of the true wind. You need to do a vector addition (or subtraction) of the two speeds, and that gives you the magnitude and direction of the apparent wind. Again, this is explained, with drawing, in most basic manuals on sailing.--

591:

understand? A team of people worked for months to do that, but further improvement is surely possible. How can you interpret the polar chart in Ref 41 in any other way than showing downwind VMG greater than wind speed? You are correct to state that the USA 17 data is no longer readily available on the 2010 Cup site. I had not realized that it had been removed. I will restore it. You will find the partial timings in older versions, for example

31: 651:

The is hardly dubious, and certainly not original. In fact, it is trivial. The videos that you refer to pertain only to the propeller-driven cart that goes dead downwind faster than the wind. The fact that boats with conventional sails, such as iceboats, sand yachts, and USA 17, can achieve downwind VMG greater than wind speed does not depend on any videos: it is documented from reliable sources, as cited in the article.--

999:; if you don't agree with it, then please update both articles so that they are consistent, and provide a reference. I like the borders on the tables, they are easier to read that way. I don't know what should be cleaned up in "Further Reading", maybe somebody can take care of that. Same regarding general improvement of the language: I've many revisions to the article, incorporating many good suggestions that were made on 1333:

strength of the apparent wind, once it goes forward of abeam, provides more forward force (and therefore power; force x speed is power) is the same as claiming perpetual motion and we know that does not work. Once the apparent wind goes forward of abeam, it is the component of the wind perpendicular to the direction of motion, which is constant, that provides all forward force and is more than adequate to explain DDWFTTW.

1486:

forward of the beam. Therefore, the apparent wind will always be aft of the beam and a spinnaker will increase the speed of the boat. However, downwind progress might still be faster if the boat gybes downwind, even with a spinnaker, and indeed most books on racing tactics say that it will usually be faster to avoid a dead downwind course and to gybe back and forth, even if the sailboat is using a spinnaker.

3756:

or the other frame. The pilot has it easier deciding his airspeed, the spectator has it easier calculating ground speed. As for the boat I want to monitor progress to a point fixed my frame of reference which is not fixed in the sailors. I'll present / I am presenting the maths. Easier for this particular purpose in my shore-based frame. Sail trimming? Easier if you're in the boat.

4325:

resistance of the APPARENT headwind generate an opposite force, so eventually the boat stops accelerating. The point is that the driving force, the one from the sails, is generated by the APPARENT wind, and that is what you have to use when doing the physics. Again, this is explained in all basic textbooks, which disuss not only trim, but also the forces and the speed vectors.--

2553:, so the speed of the ball must remain constant (Sin=Sout) in this reference frame. This makes the calculation of the change in momentum of the ball across the sail a simple trigonometric exercise involving the speed of the ball and the angle it is turned by the sail. The change in momentum (a vector) is the same as the force (also a vector) exerted by the ball on the sail. 2417:

experiment takes place in a vacuum). If that resistance from its motion along the rail is very small, then the boat will continue to advance along the rail (conservation of momentum and kinetic energy), slowing down only slightly. Then the panel is hit by a new billiard ball: this will result in the transfer of some energy and in an acceleration of the boat.

2124:

has a downwind VMG of 2x the speed of the wind, there is twice as much energy available from the surface as is required to do the needed work on the air. If anyone needs such a simple analysis to be convinced, I'm happy to produce it ... meanwhile here is a link to a very comprehensive one produced by the legendary MIT Professor of Aerodynamics Mark Drela:

4775:

the true wind, and work out the speed and force diagrams relative to the boat as it accelerates. Then, if you wish, convert them to the stationary frame of the earth. The mathematics of the boat frame are simple, they are presented in the article. The mathematics of the shore frame are more complicated, which is why you won't find them in textbooks.--

1087:. At some point,the downwind vector of the boat velocity must be equal to the wind velocity, at which point. The only remaining wind vector is perpendicular to the wind direction and is entirely due to motion of the boat. At this point and beyond, the "wind" is created entirely by the motion of the boat and thus becomes a true perpetual motion hype.-- 2023: 3537:

magnitude of the velocity component of the boat in the direction of the boat's acceleration is greater than zero. This is exactly the opposite of PBs claim, that the magnitude of the air's velocity component (in boat's acceleration direction) must be greater than the magnitude of the boat's velocity component (in boat's acceleration direction)

2599:

considering changes in NS velocity the WE velocity of the colliding objects van be disregarded. That is what vector decomposition lets us do, and it's done all the time in high school and undergraduate physics. It's not controversial and it's not rocket science. (Well, actually it is rocket science, but you get my drift, no pun intended).

1273:

course). The apparent wind will still generate a forward force component, so the boat will continue to increase in speed and the apparent wind will shift even further forward. As explained in the main article, this can result in velocity made good (progress in the direction of the wind) that is greater than the velocity of the wind.

2467:

and differential energy between the air and water that provides the capability to sail. If there is no difference in velocities (vector quantities) between the air and the water, a sailboat cannot move through the water in any direction without an external energy source (paddle, motor, etc.). But if there is no wind over the

1064:

case of an iceboat) and the resistance of the apparent headwind induced by the boat's progress. As explained in the main article, when a boat sails at an angle to the wind, it can capture energy from the wind, even if its downwind progress is faster than the wind itself. This is because of the apparent wind shift, see below.

2798:), and therefore "pushes" northward on the wind, from which Newton's Third Law plainly grants an ongoing thrust southward onto the sail, and, through the rigging, from the sail onto the yacht, more than adequate to counter all frictional forces for iceboats, land yachts, catamarans and other high-performance craft. 524:* claims that are contradicted by the prevailing view within the relevant community, or that would significantly alter mainstream assumptions, especially in science, medicine, history, politics, and biographies of living persons. This is especially true when proponents say there is a conspiracy to silence them. 3581:

viewpoint. (Otherwise that the ground is moving relative to the observer makes it more complicated - our target point would be moving too!) Having thus being allowed to discard apparent wind, where is the component of the wind pushing the vessel faster downwind than the wind itself!?!? There cannot be one.

1403:

that wind, because it didn't have any effect on the air far away from it. Anyway, Newton's laws apply in any frame. So all one has to do is apply them, and they tell us there can be a net force that acts to accelerate the boat even when the downwind vmg of the boat is 1 or greater. See my example below.

4566:

and is thus PROFOUNDLY TRUE. Nevertheless I did not invoke it. And you cannot exert a force in particular direction in collision with a 2nd object unless one is travelling faster in that directon than the 2nd object. That is why the wind cannot propel you faster than itself IN THE DIRECTION of the

4537:

Your criticism here is based on the 'conservation of velocity', which is not a recognised principle. 'Conservation of momentum' is, but as Eyytee says below , in order to 'see' momentum being conserved as a sailboat goes by, you would have to 'see' all the disturbances left behind in the air and the

4457:

Actually the math is not at all complicated in the frame of the boat (not the frame of the wind): it is elementary trigonometry and you will find it in the article. Again, you will also find it in basic texbooks on sailing. And I would appreciate it if you stopped using expressions such as "you are

4426:

No, too difficult, I'm on the shore, you want to invoke relative (or apparent) wind without doing the very complex maths that requires. You're out of order! (Different argument: I use the "set of the sail" as a device to average out the sail's curvature and drag. There is no important consequence

4200:

on the equivalence of inertial frames. I am grateful to you acknowledging this in advance as it avoids argument later. As every inertial frame is equivalent to every other I am doing the maths in the frame where the maths is easiest. No very complicated trigonometry is required im my "spectator on

3644:

Apparent wind direction and velocity is not a red herring. It is part of the physics of what allows a sail powered craft to travel faster than the wind. The sail is an airfoil, not just a flat obstruction to the wind. An airplane does not fly because the wing is tilted at an angle such that the wind

2932:

The wind force experienced by a sail is perpendicular to the set of the sail. That force is most conveniently decomposed into two forces at right angles to one another (the vector sum of which is the force on the sail). The first force is in the axis of the vessel i.e. in the rolling, ski-ing, keel

2883:

Streamlines are not some sort of hand-waving black magic invention; please stop and attempt to understand them. The vector diagram above is equivalent to moving the sail sideways and slightly down in calm air (less than 22.5 degrees South of West in the diagram, assuming South is down. In that case

1649:

If the forward component of the force on the sails is sufficient, the boat can continue to accelerate, the apparent wind will shift further forward, and VMG will be greater than 1, that is, greater than the speed of the true wind. That is, the boat will progress downwind faster than the speed of the

1272:

But if an iceboat sails downwind at an angle to the wind, there will be an apparent wind shift: the apparent wind will move forward. As explained in the main article, the iceboat will eventually find itself on a broad reach with respect to the apparent wind (apparent wind at 90 degrees to the boat's

1187:

It is correct that, if the boat is going straight downwind, then the apparent wind will be nearly zero. But that is not the point. The situation that is of interest is when the boat is NOT going straight downwind: for example when it is going at 135 degrees off the true wind. In that situation, as

1164:

QuietJohn: It was obvious from the next sentence below, and from the grammar of the sentence you tried to correct, that the original statement refers to adding the true and induced winds vectorially; the original statement is correct: the induced wind and the true wind are to be added as vectors to

994:

I have corrected the introduction and the terminology. Regarding the drawings, the arrows point in the correct directions for the vector algebra to work: just think of the case when the boat is moving directly upwind or directly downwind; I wear thick glasses, so I like the large font, and I presume

667:

sense to me, and I haven't found any data to convince me otherwise, so it's either wrong, or the page isn't providing the necessary information. Guide me to the polars- or other corroborating data and I'll be happy to concede. Until then, much of this article isn't appropriate for a Knowledge entry.--

99:

Many (including some editors) seem to have a problem understanding how achieving a downwind-VMG greater than true-wind-speed is possible. I propose to include a introductory vector diagram, that shows how the sail can produce a force with a positive forward-component that propels the boat, under this

4713:

I am leaning towards agreeing with Nigel. As all of you know, I am very patient and more than willing to try to explain admittedly counter-intuitive concepts. But, given my two years of physics at MIT (followed by a BA in Math and a PhD from Harvard), I am beginning to be a bit tired of being told

4397:

Nigel: I already provided the references, but Paul Beardsell deleted them. Bentwaithe's book explains all this (and more) in excruciating detail. And explains how high VMG is achieved. Again, all I did was summarize what is in Bentwaithe and the other cited sources. But Paul Beardsell has deleted

3880:

This is not difficult at all. It is perfectly valid to use a reference frame where where the sail is at rest in order to compute the force on the sail. The air velocity in this frame is called "apparent wind". Since the coordinate transformation doesn't involve rotation, the force vector on the sail

2771:

greater than the true wind. The angles chosen are not at all optimal but were selected to make understanding the concept as simple as possible. This model has a theoretical limit to the maximum VMG downwind so thermodynamic principles are not violated, which an important criterion in assessing any

2662:

Then someone says oh yes you can! And they're the skipper of an America's Cup yacht. Or they have endless qualifications, and they've read every book on yachting, and they've maybe even written one. Or someone has a PhD in Physics and they disagree? Well, then they're wrong, they forget the basics,

2630:

Yes, we do need a diagram, the one showing how the true wind is shifted forward and becomes an APPARENT wind that allows the boat to accelerate past the speed of the true wind. Any sailor knows about apparent wind shift and the resulting decrease or increase in apparent wind speed. You deleted the

2586:

go in the article. This is high school physics only. This could easily all be substantiated from WP articles, but there is no need, as it so uncontroversial. If you disagree with it then you are not literate in the most basic physics. Sorry. Leave aside your beliefs that sailing downwind with a

2495:

it encounters the sail that is transferred to the sail. There is another point to be made here: momentum is a vector so by definition it has a magnitude and a direction; if the direction of the momentum of a billiard ball changes but the magnitude of the momentum does not, that still constitutes a

2412:

Consequently, the direction from which the billiard balls are coming changes: they appear to come from further forward as the boat increases in speed. If you adjust the angle of the flat panel, the incoming balls will still create a forward force on the panel (in addition to the sideways force that

2083:

As stated below, it doesn't really matter what the lift/drag ratio is, so long as lift is greater than drag. According to Bethwaite, lift is about 2/3 of the total force on the sail. What matters is not the drag induced by the sail, because that is always overcome by the lift induced by the sail.

1620:

That is correct. I should have referred to the lift, which is the force on the sails net of drag. According to Bethwaite, lift is about 2/3 of the total force. But this does not change the argument, because lift and drag increase proportionally, with the square of wind speed. So there is an excess

1120:

You are correct, it is vectorially subtracted. The drawings in the article show this. You are correct that, at some point, the downwind vector of the boat velocity is equal to the wind velocity: for a specific example, see the drawings for 135 degrees below. You are correct that the apparent wind

792:

I'm not sure what you mean by "continous uninterrupted progress". Hardly any boat progresses at uniform speed. Variations in the strength of the wind and waves cause any boat to advance at varying speed. As you say, the wind-and-anchor driven device that you describe is not particuarly surprising

590:

You appear to be starting from the assumption that VMG greater than wind speed cannot be achieved, and thus propose to delete various bits. How about starting from the assumption that VMG greater than wind speed can be achieved, and then proposing ways to clarify the article so that it is easier to

558:

In fact, several sections in the article - The last 2 paragraphs and table in Sailing perpendicular to the wind are erroneous and speculative. Sailing on a broad reach, velocity made good are similarly speculative and unsupported. Aerodynamic drag on the sail is completely ignored. Boat speed is NOT

386:

It is not Physics 101, and you have not provided a citation to show that it is. Worse, you have added to the main article the statement “Nevertheless this remains slower than the time taken for a free floating balloon to travel the downwind track directly”. This is Original Research, not supported

4774:

Of course the frames are equivalent, but the easy frame is the boat's frame: that is why all textbooks use that frame. The shore frame is hard to get right, and that may explain why you are having trouble seeing this. Please try the following: imagine an iceboat that is sailing at 135 degrees off

4481:

Paul Beardsell is correct in saying that the frame does not matter. However, the vector algebra is easier in some frames than in others. The deleted material used the frames that are adopted by all textbooks, precisely because the presentation is simpler. I still cannot understand why Paul refuses

3845:

is not the most helpful BUT one has to decide what one's coordinate system is and then stick with it. By changing one's frame of reference one can become disoriented and start believing three impossible things before breakfast. If one chooses one's frame of reference to be the vessel then analysis

3755:

Standing at the side of the runway I see the true wind, I see the speed of the a/c relative me. I add/subtract the two. So what? Both POVs are equivalent, the pilot's and the spectator's. What happens in reality is the same in both frames. some things are easier to measure and calculate in one

3536:

I have a very simple counter example that disproves PB's statement at the top of this section (in its current form) : Boat is moving perpendicularly to the true wind, and accelerates forward. The magnitude of the velocity component of the air in the direction of the boat's acceleration is zero. The

3480:

I have sent you the extracts by E-Mail on the morning of 9 June. If you have not received them, then please send me an E-Mail to that effect. Re USA 17, the mark was dead downwind when the race started, and the wind shifts were not significant. So, for all practical purposes, the VMG to the mark

3024:

Nobody here claimed that tacking upwind contradicts classical mechanics. It just contradicts your initial statement on the top of this section. On an upwind tack the air doesn't have a velocity component in the direction of the boat's acceleration which is greater than the boat's velocity component

2951:

This is correct. And if you work out the force diagrams for an iceyacht sailing at 135 degrees of the true wind, you will see why it can accelerate to 6 times the speed of the wind, and thus achieve VMG far greater than wind speed. I will work out those diagrams and publish them here, but it will

2466:

As noted above, the billiard billiard ball model is a momentum- and energy-transfer model. That is, generally speaking energy and momentum are transferred between the two media - air and water (or air and land) - and anything those media touch. To be more accurate, it is the differential momentum

2401:

One can think of the molecules of air as billiard balls, and the sail as a flat panel that is struck by the billiard balls. There is a transfer of kinetic energy and momentum from the balls to the flat panel. If there is no contsraint on the panel, it will wind up moving in the same direction and

2062:

etc. would provide a more convincing treatment. Given that wing angles of attack can be as low as 5 degrees and lift/drag ratios can be as much as 10, the ideas presented here seem feasible. However, the analysis presented seems dubious. I also remember my math teacher proving that 1 = 2. What I've

1675:

Thus, the forward force when the iceboat is close hauled will be 38% of the total force on the sail. Again, whether or not that is sufficient to accelerate the boat further will depend on the resistance caused by the apparent wind (the headwind induced by the speed of the boat). As stated earlier,

1587:

Consider an iceboat that is sailing at a course that is 135 degrees off the true wind. At the beginning, the boat is stationary. We will use the value "1" for the speed of the true wind. The apparent wind is equal to the true wind and comes from the same direction as the true wind. The velocity

1558:

As promised, I've developed force diagrams that are meant to make the situation easier to understand. But I'm not convinced that what I have done can be understood easily. At first, I thought of making three diagrams, showing velocities and forces for three situations: (1) standstill (2) apparent

1446:

QuietJohn is confusing induced wind (created by the boat) with apparent wind (sum of true wind and induced wind). However, original article is also a bit fuzzy/careless. First: the boat is on a beam, not a broad, reach, wrt the apparent wind when that wind is perpendicular to the boat's motion.

1141:

Gautier lebon: this - "a sailboat uses energy from the wind to create an apparent wind, which allows it to obtain even more energy from the wind." - is misleading in general and is incorrect even after the apparent wind direction goes forward of abeam. See discussion below about the perpendicular

1063:

If a boat can capture some energy from the wind, then it can use that energy to propell itself downwind faster than the wind. There is no violation of the law of conservation of energy because the energy captured from the wind is used to overcome the resistance of the surface (insignificant in the

666:

ALmost all of the citations I have checked have been dubious or inaccessible to me. If the NALSA site (which I have looked at and searched for relevant data) has the supporting data, please CITE the web pages that support your claims. I'm quite neutral about the DDWFTTW phenomenon. It doesn't make

4585:

Once again, you fail to undertand that the only relevant wind is the APPARENT wind, not the true wind. Have you actually ever sailed a boat? All sailors know that sails are trimmed with respect to the apparent wind, not with respect to the true wind. And that it is the force of the apparent wind

4546:

in that wind due to the angle between its movement and the real wind. Just like in the vector diagrams in the article. One thing that's weird is that, unusually, the wind due to the boat's movement is much larger than the real wind. The real wind mostly serves to add a small angle to the 'movement

3673:

Very much a side issue but your understanding of aircraft flight is also limited! Whatever the mechanism, Bernoulli pressure differnces, whatever, those are one step removed from the basics. Newton still applies. To exert a force upwards a force force downwards is required (Newton 3: every force

2859:

in the actual case it is obvious that the sail is not perpendicular to the true wind direction, but is instead angled between the boat's velocity vector and the vector perpendicular to the true wind. That angle means that relative to the streamlines of the wind moving south at the true windspeed,

2542:

In a typical apparent wind diagram of a sail, the billiard ball (apparent wind) comes into the sail at some angle of attack off the centerline of the boat and leaves the sail going parallel to the centerline. Because the sail is not moving in this reference frame it can neither give energy to nor

2526:

Other important concepts are the conservation of momentum and energy in any inertial reference frame. If a quantity of momentum is extracted (subtracted in a vector sense) from the wind, that same quantity of momentum must be added (again, in a vector sense) to the boat or to the water; note also

2123:

Additionally, one needn't even get into particular L/D ratios to mathmatically demonstrate that no physical laws are violated -- one merely need to do a simple energy balance. Using the simplest of 'force * distance' formulas one can easily see that in any tailwind, when the vehicle (any of them)

1664:

The chart below shows the forward force resulting from the apparent wind when the iceboat is at a standstill (V0), on a broad reach (V1) and close hauled (V2). The force generated by the wind is perpendicular to the sails. It can be seen from elementary trigonometry that the forward component of

1653:

The chart below shows the apparent wind and the sails on the iceboat when it is at a standstill (V0), on a broad reach (V1) and close hauled (V2). The angle of the sail is half of the angle of the apparent wind, which is the optimal angle in terms of generating a forward driving force. The force

1359:

I agree that whether or not the forward force increases is not particularly important for this discussion. What matters is whether the forward force (net of the drag of the sail itself) is sufficient to overcome the resistance of the surface and the resistance caused by the induced headwind. But

1309:

Well, it must be true that it should be explained more clearly: I don't know how to do that, help would be appreciated. You are correct that the apparent wind shifting forward reduces the forward force component. BUT the apparent wind become stronger as it shifts forward, so the resulting forwar

1214:

This is getting silly. All three of us are saying to add vectors. Where is the disagreement? Nobody is saying that subtracting magnitudes is correct for the general case, or is correct for any case but straight downwind. The current case, which QuickJohn mistakenly tried to correct but which I

715:

I disagree that the material in this article is in conflict with wiki guidelines. The material discussed here is standard, well-known to any expert on the physics of sailing or wind power and to many informed amateurs. The fact that some people find it non-intuitive or have trouble accessing the

650:

The cited book by Bethwaithe is a perfectly reliable, high-quality source, that summarizes years of data and developments. The NALSA web site contains many references and data that show that iceboats routinely go downwind at VMG greater than wind speed, and it also explains why this is possible.

4661:

so you can draw your diagrams carefully and correctly this time; personally I look forward to those diagrams because we all learn from seeing different perspectives on any topic. Once that is done, note that there are recorded on this page a very many wrong things you have stated not only with a

4370:

of sail-cloth has a component at a right-angle to the square inch, and another (hopefully much smaller one) roughly backwards along the cloth due to drag. The curve or 'belly' near the front of a close-hauled sail (the luff) contains lots of square inches facing almost forwards, and it is largely

2482:

One thing to note in Gabon's example above is that the flat panel (sail) is perpendicular to the true wind direction. In that case the sail and boat VMGDW (Velocity Made Good DownWind i.e. parallel to the true wind) cannot exceed the true windspeed. Mr. Beardsell's analysis (Physical mechanics,

2144:

Also, if folk are looking for for more math, here is a link to another extensive treatment by MIT Professor Mark Drela on the topic. His treatment is specifically aimed at a DDWFTTW water craft, but near the end he addresses a ground based wheeled vehicle and makes the following statement "This

1485:

Most sailboats cannot accelerate enough when they sail downwind so that the apparent wind shifts to come forward of the beam, and this because the resistance of the water is so large. That is, the resistance from the hull prevents the boat from moving fast enough so that the apparent wind shifts

1402:

As for the boat "creating the apparent wind", not really. The apparent wind is just the average speed of air molecules in a reference frame in which the boat is at rest (just as the "true" wind is the average air speed in a frame in which the land is at rest). The boat obviously didn't "create"

887:

Thank you for your edits. Regarding the section in question, please specify what exactly you are objecting to. There was NO disagreement regarding the theoretical possibility. There was skepticism regarding whether or not the propeller-driven cart was was a hoax or not. You (and others) state

769:

I've posted some responses in talk:sailing. Another interesting topic is sailing directly into the wind. I've never seen a turbine / prop craft "work" directly upwind in what I consider to be a controlled test - the so-called proof has been less than convincing. The "spork33" device can be simply

312:

At the point where (speed over the ground)/cos(angle to the true wind)=(speed of the true wind) the sail provides no forward force (i.e. in the direction of the keel / wheels / skates). The slightest friction brings the vessel gradually to a halt. But, if all friction is eliminated the downwind

118:

Your diagram is very good! Also the two force diagrams at the bottom are illuminating. Maybe you could add the lateral force of the water acting on the keel/hull/dagger board balancing the large lateral component of the sail force, which is essential for the whole argument (without it the hot-air

4679:

OK, I'm outa here. If you are going to start rearranging other people's helpful comments so that they make no sense any more, and flooding the page with your own disordered comments, and getting abusive ("physics-illiterate" indeed!) and SHOUTING, then you can continue to argue without me. After

3712:

Paul H has it right. An aircraft's wing sees only the apparent wind. That is why you don't want to take off with a tail wind: it would take too long to accelerate enough so that the apparent wind allows you to take off. All pilots are well aware of the fact that only the apparent wind counts,

687:

shows downwind VMG greater than windspeed and it is consistent with the polar charts published by Bethwaite. As stated many times on this talk page, and in the article, the published data from the first regatta of the 2010 America's Cup shown that USA 17's downwind VMG was far in excess of wind

4324:

Sails are trimmed with respect to the apparent wind in order to optimize the force generated by the APPARENT wind. The sails generate a force, which drives the boat (technically, the force causes an acceleration, which results in an increase in velocity). The friction from the surface and the

4067:

Easier is to choose one's frame of reference as the ground. Then the boat moves and everything else remains constant! Wind relative to the motion of the boat remains important to those IN THE BOAT but unimportant to those in OUR frame of reference. Physics (in particular certain conservation

2462:

I have alway preferred this model, it is easier to understand and makes much more sense than the reams of bogus verbiage I have seen over the years invoking Bernoulli and phantom venturis. I used it yesterday to explain all points of sail to a neophyte in about ten minutes. I will try here to

1591:

The boat will accelerate and will eventually reach a speed equal to to .707 times the speed of the wind. At this point, the apparent wind will come from 90 degrees (broad reach) and the speed of the apparent wind will also be .707. VMG is .5 This is shown by the value "V1" in the chart below.

748:

I know that this section seems hard to believe. Please read the citations and study them carefully before posting comments to the effect that this section is obviously wrong. I too did not believe that this was possible, until I did the research and found the citations given in this section.

3580:

Apparent wind direction is usually a red herring in this type of discussion. It is useful if you want to explain what is happening from the perspective of someone IN the boat. No, as physics allows me, I will continue to look at this from the stationary (in comparison to the ground) spectator

2408:

Now imagine that the flat panel (the sails) and the boat are constrained to travel at an angle (say 135 degrees) to the direction from which the billiard balls are coming. for example because it is mounted on a rail. When the balls hit the panel, there will be a transfer of kinetic energy and

1332:

There is no increase in forward force as the apparent wind goes forward of abeam. The apparent wind getting stronger as the boatspeed increases cannot not make more forward force available (for fixed-wing sails). Neither is the concept needed to explain DDWFTTW. Claiming that the increasing

584:

from the 5th sentence of paragraph 5, beginning "Again, simple trigonometry can be used to calculate the velocity made good." should be removed. I didn't find any references to AC course times in any of the AC references (32, 34 - 38). Some of those pages are now gone - therefore no VERIFIABLE

248:

Mr. Beardsell: if I understand that statement as written, then what you are saying is equivalent to saying that the maximum (speed over the ground) = cos(angle to the true wind) * (speed of the true wind). Am I correct that that is indeed what you are saying? At what points of sail do these

3461:

No USA17 did not! It had a VMG to the mark but the mark was NOT directly downwind! This point has been made to you already by me and on another article's talk page by someone else. You did say you were sending me extract's from B's book. I have not received them. But no matter, I am sure.

2613:

The question then is, and you should all have seen this coming, if the yacht is going faster than the air in a southerly direction, how can collision with the air accelerate it in a southerly direction? And, sorry to disappoint, that's a rhetorical question, there is no satisfactory answer.

2416:

Now imagine that the billiard balls are relatively spaced out, so that one hits the panel only every second. In between billiard balls, the boat will be slowed down by the resistance of its motion along the rail (assuming that the resistance from the air is negligle, for example because this

1245:

Yes, I am probably confused about who is saying what about what. Maybe it would be better if you started a new section? I'm no longer sure about what QuickJohn is arguing, apart from the fact that, according to him, the citations are insufficient or insufficiently reliable: he has made that

3166:-6 (if you find that patronising I apologise but there are lots of things that simple which I am having to explain), which is why sailing towards the wind presents no fundamental problem in my argument. It is DOWNWIND TO A POINT QUICKER THAN A BALLOON which is impossible. Examples in prep. 2598:

have had a southerly component to your velocity of more than 10mph. Otherwise the collision would have slowed me down in that particular direction. Uncontroversial, and of course we can decompose any particular motion so that it consists of the sum of one NS vector and one WE vector. When

1277:

The conclusion here is correct but the explanation is faulty: the apparent wind shifting forward must *reduce* the forward force component. It cannot be otherwise. It is possible that is not what is meant here; if so it should be written more clearly. See discussion below about component

540:

This "issue" has been publicized since at least 2008 with no peer-reviewed publications of such a spectacularly counter-intuitive notion. It is VERY hard to believe that such a scientific breakthrough could remain an orphan with no better explanation than a very dubious "thought experiment".

287:

It's like when something is incorrectly spelt in English. The literate ones don't need to be told the correct spelling. Those mathematically and physically illiterate cannot be explained to, and those that are don't need to be told. But, if you need a few pointers (ha!), start by properly

3445:

Velocity = speed. Downwind VMG = progress over the water (or ground) towards the downwind mark = cos(ang) time velocity. USA 17 had VMG of 19 knots in 5-10 knots of wind. That is evidence. The models and explanations are found in the material that you deleted, which was all supported by

2522:

wind determines the actual momentum transferred. Intuition may say the best performance has to do with making use of the momentum of the wind in the direction it is blowing, but that is not always the case. If the sail can be configured to push (change the momentum of) the wind in a given

4365:

you will see that the Bernoulli explanation of lift is now discredited. And anyway, thin cloth sails are so easy to understand, we do not need it. Paul B is nearly correct saying that, "The wind force experienced by a sail is perpendicular to the set of the sail". The force due to the wind

2106:

Failing to see a division by zero, and acknowledging the possibility of a lift/drag ratio of 10, I'm not sure what the problem is. As long as the airspeed shown in the diagram is in the correct direction, and the proportionality of airspeed to force is not negative (the coefficients are :

1997:

If an iceboat has a sail area of 6 square meters, and if it is sailing at 5 times the speed of the wind, the the foward force will be about 6*1.54 = 9, assuming the lift cofficient is 1. But the lift cofficient is about .66 (according to Bethwaite), so the forward force will be about 6.

1638:

So the boat will continue to accelerate and will reach a speed equal to 1.41 times the speed of the wind. At this point, the apparent wind will come from 45 degrees (close hauled) and the speed of the apparent wind will be 1. VMG is 1. This is shown by the value V2 in the chart below.

1601:

Unfortunately, not true. Wings (and sails) come with a penalty called drag in addition to lift. Lift would typically be perpendicular to airflow and drag in the same direction as airflow. So, in addition to runner resistance, you have aerodynamic drag which acts to slow the boat down.

721:

ws). Peer review is going to be hard to find, because no one would bother (or be able to) publish an article in a scientific journal on something this basic. It would be rejected by the editors (before the peer review stage) as too obvious and not of sufficient scientific interest.

2764:

The actual steady-state numbers will depend on the actual configuration (sail, hull type - iceboat, land-yacht, etc), but the diagram above clearly demonstrates that there can be a southward force of a southerly true wind on the fixed sail of a craft moving across that wind with a

3518:

Yes, we understand what you are saying. The problem is that published polar charts show the opposite, as does data from USA 17, despite your skepticism (the wind shifts were nowhere near big enough to counter the fact that USA 17 had a downwind VMG of 19 knots in 5-10 knots of

1520:

The citations in the article include many web sites, which are easily accessible, and a published book (by Bethwaite) which is still in print and easily available. One of the contributors to this discussion has provided a second book, also still in print and easily available:

4030:

and I believe it will be difficult for readers to follow the maths. I am simply using another equally valid frame, as you acknowledge. And, as I am sure you must know, the RESULTS calculated in one frame are the same as those calculated in every other frame - that is what

2026:

showing exactly how this works, given drag sufficiently small (doesn't need to be zero). What is appealing about these types of disputes over religious or political, is that they should be able to be resolved very quickly around a chalkboard. I suggest we attempt to do

2009:

iceboat has a sail area of 6.5 square meters and can achieve 5 times the speed of the wind. This is perfectly consistent with the calculations above, given that the aerodynamically equivalent frontal surface of the iceboat is indeed probably about .33 square meters.

508:

First and foremost, there really don't seem to be any RELIABLE, VERIFIABLE sources. Secondly, what sources there are appear to be reports of original research, none of which has made it through a peer review process since they first appeared in November of 2008.

1423:

The apparent wind is not just resulting from the motion of the boat but also from the true wind. Apparent wind is the air velocity relative to the airfoil, and it is the standard approach to use that relative velocity to calculate the force on an airfoil based on

3192:

Don't spend too much time on it, because sail-powered vehicles CAN get to a point downwind faster than a balloon. Are you saying you don't believe yachts like USA-17 and almost any iceboat or land-yacht can't achieve VMG downwind at a multiple of the true wind

4751:

Anyone with any physics (I know you have some) will tell you that all inertial frames of reference are equivalent. If you do the analysis from the POV of the sailor in the boat you will need to cope with his measurements of the wind, speed and direction, the

1670: 1215:

and others responded to, *IS* for sailing straight downwind; look at the next sentence" "If an iceboat sails dead downwind, ..." I agree the interesting case is later and not straight downwind, but that is not the case where QuickJohn made his comment above.

3241:

The term on the left of the inequality is component of the yacht's velocity directly downwind, the term to the right is that of the balloon. If the inequality is not true then there is no possible impulse of the wind on the yacht in the windward direction.

4289:. You need to review your physics. In your sailing books they will be talking about sail trim, they talk about the POV of the sailor. I use an alternative easier but equivalent frame of ref. Equivalent. Please think again. 08:19, 9 June 2010 (UTC) 3559:

is making a fundamental error in the above analysis. He assumes that the boat interacts only with the air mass. He neglects the interaction with the water. While in reality the resulting acceleration of a boat depends on both forces(keel and sail), and

1310:

force component is still signficiant: see the calculations below for the iceboat sailing at 135 degrees off the true wind. I will work out the actual forward force component and post that on this discussion page, but it will take me a couple of days.--

229:

There is none. The forward force on the vessel disappears when (speed over the ground)/cos(angle to the true wind)=(speed of the true wind). That's the argument in a nut shell. Impossible. And any CORRECTLY drawn vector diagram makes this plain.

1644: 3231:

of the wind I have done so myself! But the ice yacht travelling at 3x windspeed over the ice does so at such a shallow angle that it does not get to the downwind point before the balloon. The ice/land/sea yacht is constrained thus, downwind:

1659: 1101:

Sorry, you are confusing velocities with energy. To make a "perpetual motion" claim, you have to show that energy would not be conserved. It is not sufficient just to state, that something about the velocity vectors seems counter-intuitive to

288:

labelling everything, say what is being decomposed into what, make the lengths proportionate to the speeds, if that is what you're drawing, or forces, if that is what your drawing. You're strumming on a guitar without having even tuned it.

2001:

The drag component will be about 18. So the speed of 5 times the speed of the wind can be achieved if the frontal surface that creates the drag (the surface of the mast, hull, runners, etc.) is less than 0.33 square meters (18*.33 = 6).

593:]. The "no hull speed" limitation is not speculative, it is the actual situation of an iceboat. It is easier to understand this stuff if you think of an iceboat. Then to transpose that to a high-performance catamaran or planing hull.-- 2587:

VMG towards the directly downwind mark is possible. Leave aside what you've read the skipper of Oracle in the AC as having said. Just stay with the physics. I write this just to try and shake your convictions ever so slightly.

2378:

The energy that drives a boat comes from the transfer of kinetic energy from the wind to the boat's sails: the molecules of air strike the sail, a force is induced, and momentum and energy are transferred from the air to the sail.

3288:(parallel vectors pointing in the same direction) has been achieved by many crafts, not least USA-17! That's an empirical fact, and there are models ('explanations' if you want) in the framework of classical Newtonian mechanics. 1056:

A balloon drifts downwind at the speed of the wind. It uses no energy to do that. Similarly, an iceboat can drift dead downwind at very close to the speed of the wind, because the friction of its runners on the ice is negligle.

2413:

has to be resisted by the rail on which the panel is mounted). There is still a transfer of energy from the balls to the panel, and the boat may or may not accelerate, depending on the resistance to its motion along the rail.

3499:

VMG <= boat-speed. Where the desired point is directly downwind, VMG <= windspeed. Note we talking about the component of the boats velocity directly downwind, not its speed over the ground. Have I said that before?

2478:

Ultimately, the existence of differential momentum and energy between the two media are the reason it is possible to move in any direction relative to the true wind, while using that same true wind as the sole motive force.

3100:

Give me just a little slack. It would have been technically more correct to say "greater in magnitude" than "faster", but I suspect you did know that's what I meant. I will make that adjustment to my original text, thanks.

3069:

Then please explain what you mean by "faster" without invoking scalar speeds, when you say "velocity component in that direction faster than the speed of the yacht in that particular direction"? What do you compare and how?

4161:

seems to deny basic physical principles like the equivalence of inertial reference frames, by suggesting the Earth's frame is the only valid one, and generally denying the validity of analysis which uses a different frame.

3621:

What is also elementary is that the analysis can be done from a spectator's POV, one who is not in the boat and one who does not experience apparent wind. That is what is meant by the equivalence of frames of reference.

3991:

You on the other hand suggest, that we should compute the force on the airfoil WITHOUT using the air velocity relative to the airfoil(apparent wind). I will force you to do the maths if you insist on this non-standard

697:. So you have multiple sources providing data that confirm the physics explained in the article. And that physics is not Original Research, it is simply a summary of the physics found in most text books on sailing.-- 4624:) while still having air particles moving at the true wind velocity colliding with the windward side of the sail of that same boat. Before you reflexively post "impossible" consider the implications of the following: 3802:

Well, if you actually do work it out, you will see that it is far simpler in the frame of the boat. The same hold for planes: pilots look at the relative wind speed, not speed over the ground plus or minus true wind

2702:. It is difficult to explain a satisfactory answer when using a reference frame fixed to the water, but here it is (using the reference frame moving with the boat makes the answer obvious). (removed some snarkiness 2084:

What matters is the drag induced by (1) the surface (insignificant for an iceboat) and (2) the windage on the mast, hull, etc. It is that drag (the drag from the induced headwind) that limits the ultimate speed.--

328:

This statement by Paul Beardsell is Original Research. It is not supported by any citation. It is also incorrect, as shown by the citations that Paul deleted from the original article. I will post more on this

1539:

As explained below, the material in the article is not sufficiently original or suprising to be published in a peer-reviewed scientific journal. So it is not clear what additional citations should be provided.--

347:

It isn't OR. It's Physics 101. And even if it were OR, we are on a Talk page, and we can write here what is NOT allowed in the encyclopedia pages. I will not allow OR there. And it is not me who wants to put

4201:

the shore" frame of reference: I do not even have to calculate apparent wind. Much easier. But I am glad you acknowledge the frames' equivalence as then you will not be able to come back to me and say "but, the

3953:

Just to clarify: I'm not suggesting using the sail's frame for the entire analysis, just for the calculation of the force on the sail. This is the standard approach to calculate the force on an airfoil based on

716:

references isn't relevant. There are many published sources, although I think the article is sufficiently referenced as it stands now. If you want a published book reference for ddwfttw, this one is very good

2108:

0), then the accelerating force is pointing in the right direction. Finally, if it is possible that the accelerating force is greater than the drag force, which it is, we have a boat that is accelerating past

1676:

in practice it has been found that the force is indeed sufficient to further accelerate the iceboat (and in fact also other high-performance boats, including sailboats, as explained elsewhere in the article).

2543:

receive energy from the billiard ball, so the KE of the ball cannot change (again, ignoring frictional losses for now). Solving for the speeds of the ball coming in and out of its interaction with the sail:

433:

It seems many of the rest of you cannot do force diagrams properly, and seem to want to disregard 2nd law of thermodynamics, as if science is not a matter of fact but merely what you would like to believe.

149:

There is no need for that (nice as it would be). All they need show is that there is a forward component of the force on the sail in conditions where the free floating balloon is being beaten to downwind.

1399:

It's motion that lasts as long as the wind does, just like sailing slower than the wind. If the wind dies, it stops. So it's as "perpetual" as the wind, or a windmill, or any other wind-powered device.

566:

from paragraph 5, beginning "If hull speed is not a limiting factor, and if the strength of the wind is sufficient to overcome the surface resistance....." should be removed - erroneous and speculative.

2420:

So the boat will continue to accelerate until the energy lost due to the resistance of its motion along the rail is greater than the energy transferred to it when a billiard ball hits the flat panel.

3227:(not speed) towards the desired point, which in our case, is the point directly downwind, the point towards which the balloon is drifting. As to having travelled on a downwind tack faster than the 2409:

momentum, but now the panel does not wind up moving dead downwind, it winds up moving along the rail, that is at an angle with respect to the direction from which the billiard balls are coming.

616:

I don't currently have library access to the book, "High Performance Sailing". These citations need to be checked and perhaps verified against second sources - even book authors can be wrong!

2423:

The boat's ultimate speed depends on the friction from the surface, on how efficiently it captures the energy from the billiard balls, and on how much energy there is in the billiard balls.

2578:

but no, that's giving me airs and graces. This is plain classical mechanics, unadorned. It's fact, plain and simple. There is no need to put this in the article, this below is the reason

2471:, a sailboat can move on a river with current because of the differential momentum and energy between the still air and the moving water. And now that DDWFTTW has been demonstrated, it is 1385:

The apparent wind is created by the boat. So what this sentence says is that the motion of the boat creates the apparent wind which then speeds up the boat. Once again, perpetual motion.--

2904:"So when a particle of the air collides with my yacht and the yacht is accelerated in a particular direction, the particle of the air must have had a velocity component in that direction 2602:

So when a particle of the air collides with my yacht and the yacht is accelerated in a particular direction, the particle of the air must have had a velocity component in that direction

1559:

wind at 90 degrees (3) apparent wind at 45 degrees. But then I decided to make separate diagrams for the velocities and the forces, with each diagram showing the three situations.--

3674:

etc). Whatever the mechanism, whatever the meta-explanation, airfoil lift RESULTS from the directing of the airflow downwards. Flight is not a special case: Newton applies.

3962:. And this is not confusing because as I said: The coordinate transformation doesn't involve rotation, so the force vector in the ground frame is the same as in the sail's frame. 249:

statements hold true? At what points of sail do they not hold true? Could you check your original formulation to see if you meant to multiply by the cosine instead of divide?

1079:

As explained in the main article, the forward motion of the boat induces a wind that must be added to the true wind in order to find that apparent wind that strikes the sails.

1669: 205:

I never said that speeds faster than true wind velocity were impossible. And that isn't a proper vector diagram showing proper decomposition of forces. It's just a sketch

4695:

Play the ball. I moved your critique to a subsection to allow BETTER addressing of your points. Sorry about Newton 1, but that was your error not mine. Heat, kitchen etc

2483:

above) is an excellent description of why. The reason I bring this up is to make the point that when momentum transfer occurs, momentum must be conserved, but it is the

2263: 2590:

In order for one object to propel another in a given direction (consider two moving balls, or whatever) the first must have a component of it's motion in that direction

1432:. If you suggest, that we should compute the force on the airfoil WITHOUT using the air velocity relative to the airfoil (apparent wind), then I want to see your maths. 1573:

I would much appreciate comments on how to make the text below, and the diagrams, easy enough to understand so that they could be incorporated into the main article.--

578:

should be removed - erroneous and speculative. Once again makes the assumption that boat speed is a simple vector of apparent wind speed and ignores aerodynamic drag.

1643: 1498:. The winning yacht, USA 17, completed the 40 nautical mile course (20 miles dead upwind, 20 miles dead downwind) in 2h32, in 5-10 knots of wind, see for example 1022:

If you think that anything stated in the article is impossible, please consider that the following two reasons have already been stated, and shown not to apply:--

4547:

wind', an angle that this very efficient boat can continue to claw itself along by, close hauled, getting just enough drive to overcome its minimal hull-drag. --

2193: 2189: 2147: 2125: 2496:

change in momentum transferred from the billiard ball to the sail which changed it. Counter to intuition, the direction and magnitude of the momentum of the

684:

The Bethwaite book is easily available. The fact that you do not have it does not mean that it cannot be used as a citation in Knowledge. The polar chart at

276:

is NOT CORRECTLY drawn. Yet he fails to explain what exactly is not correct in the diagram. He merely makes assertions without providing proof or references.

1658: 4567:

wind. I.e., to the physics-illiterate: that is why downwind sailing to the directly downwind point quicker than the free floating balloon is impossible.

2046:

or drag) are not directly proportional to airspeed vectors. There are coefficients involved, one for lift and a different one for drag. A consideration of

995:

that others will too, since it does not make the drawings any larger. Regarding performance of normal cruising boats, I just took the figure given in the

4035:

means in this context. If the boat does not beat the balloon in my "spectator on the shore" frame it does not do so in your "sailor on the boat" frame.

1269:

If an iceboat sails dead downwind, then it will soon reach a speed close to the speed of the wind and the apparent wind on the sails will be nearly zero.

2267: 3599:

Apparent wind is NOT a red herring. The sail sees only apparent wind and can react only to the apparent wind. This is elementary Newtonian physics.--

2356:

V=a*t, meaning that an acceleration a applied for time t will increase velocity by V. If there is no acceleration, then the velocity does not change.

4515:

I have sent you the extracts by E-Mail on 9 June at 8:38 European time. If you did not receive them, then please send me an E-Mail to that effect.--

1381:

Note the key point: what drives the boat is the apparent wind, that is what the sails "see" and what the sails react to and what propelles the boat.

86: 81: 76: 64: 59: 4126:

This means that you fail to understand the basic physical principles involved. Which makes it difficult to see how to continue the discussion.--

1684:

Here is the table that I've come up with for a course that is 135 degrees off the true wind. Please check the formulas and the calculations.--

1045:

4) There isn't any easily accessible published data that supports the statements that boats can achieved downwind VMG greater than wind speed.

1067:

Also, it is easy to see that a device can be constructed that can capture the energy from the wind even when moving dead downwind, see below

2853:

If the sail were perpendicular to the true wind, then Mr. Beardsell's analysis above would of course hold (you can't beat simple physics).

2536:

The kinetic energy (KE) of a billiard ball is half the product of its mass and the square of its speed (S = the magnitude of its velocity):

1039:

2) Velocity made good downwind cannot exceed the speed of the wind because there is no force on the sails when the boat reaches wind speed.

777: 2700:

if the yacht is going faster than the air in a southerly direction, how can collision with the air accelerate it in a southerly direction?

2338:

I have the impression that some people are, at times, confusing some basic concepts. So here is a reminder of what we are talking about.--

3915:

It is perfectly valid but I will force you to do the maths if you insist on using that frame and you will find the trig more complicated

1294: 2530:

The momentum (M) of a billiard ball (our analog for the wind) is the product of its mass (m) a scalar, and its velocity (V, a vector):

2807: 2698:

The only thing wrong in what was written above here is the claim that there is, and can be, no satisfactory answer to this question:

2720: 1068: 908:

The drawings are nice but have a too large font size and the arrow indicator on the wind vectors are pointing the wrong direction.

861:

is well aware of. The section should not be added unless a reliable source is referred to (not likely as it is impossible to sail

2353:

F=m*a, meaning that a force applied to a mass will produce an acceleration. If there is no force, then there is no accelaration.

1596:

At this point, the only resistance to forward motion is the friction of the iceboat's runners on the ice. But that is negligible.

544:

As indicated in the Knowledge recommendations for such unauthenticated exceptional claims, this section should not be included.

4680:

you're done, we'll restore the page to a consensus version, but this is not helpful to improving the article at the moment. --

2148:

http://www.boatdesign.net/forums/attachments/propulsion/28167d1231128492-ddwfttw-directly-downwind-faster-than-wind-ddw2.pdf

2126:

http://www.boatdesign.net/forums/attachments/propulsion/28168d1231128492-ddwfttw-directly-downwind-faster-than-wind-ddwe.pdf

456:

It might be the other way around, given that all text books support the force diagrams that Paul Beardsell is criticizing.--

4586:

that propels the boat. That's why it is so slow to go dead downwind with square sails: the apparent wind is pretty weak.--

1494:

The article provides citations to various observed data. The most accessible data are those from the first regatta of the

4736:

Correct. But what counts is the APPARENT wind, not the true wind. That was explained in the material that you deleted.--

47: 17: 3446:

citations, in particular to Bentwaithe's book. Benthwaithe provides polar charts showing VMG greater than wind speed.--

2289: 2213: 2594:. If I am travelling south at 10mph and you collide with me and I end up travelling south at more than 10mph then you 924:

Normal cruising boats yachts can sail at a about 45 degrees off the apparent wind (50 to 60 degrees off the true wind).

134:

Good start, but the force diagram needs a drag vector, so the true boat velocity can be derived, not just a given...--

38: 1036:

1) Velocity made good downwind cannot exceed the speed of the wind because of the laws of conservation of energy

635:

No problems with sailing faster than the wind - just with a downwind velocity made good greater than windspeed.--

4205:

wind is different" as you will acknowledge, in my equivalent frame, there is no apparent wind, only true wind.

2866:

directly south along that streamline. You can't beat simple physics, especially when they are done correctly.

692: 4563: 2154: 2132: 971:

OK, that seems sensible to me. I'm busy with other things this week, but I will try to work on it next week.--

781: 750: 4780: 4741: 4719: 4667: 4610: 4591: 4520: 4487: 4463: 4403: 4330: 4248: 4131: 3808: 3718: 3604: 3524: 3486: 3451: 3421: 3005: 2957: 2889: 2874: 2844: 2811: 2803: 2777: 2755: 2707: 2683: 2636: 2619: 2560: 2452: 2438: 2392: 2343: 2317: 2241: 2171: 2089: 1689: 1654:

generated by the apparent wind on the sail is proportional to the square of the speed of the apparent wind.

1626: 1578: 1564: 1544: 1530: 1506: 1470: 1452: 1365: 1338: 1315: 1290: 1251: 1220: 1193: 1170: 1147: 1126: 1027: 1008: 976: 893: 836: 817: 798: 760: 753: 702: 656: 625: 598: 461: 392: 334: 254: 1408: 727: 717: 160:

shows. A forward component of the force on the sail while going with a downwind VMG = 1.5 true wind speed.

3293: 2150: 2128: 1621:

of lift over drag, and there is still a forward force which will accelerate the boat past a beam reach.--

124: 3366:

You and others allude to this evidence and then fail to produce it. This does not surprise me: One can

2916:

If the above paragraph was true boats could not tack UP wind, which is an observed fact for centuries.(

360:. Here, we can talk and argue but EVEN IF WE REACH A CONSENSUS what goes on the page itself has to be 4562:

No, I never said "conservation of velocity". Yet you are wrong to seemingly ridicule the term, it is

3000:

No, it won't, because I have already done them by hand. What will take time is to make them pretty.--

2614:

Therefore sailing downwind VMG faster than the air is impossible. (I removed Mr. Beardsell's QED here

2402:

the same speed as the billiard balls: this is drifting dead downwind at nearly the speed of the wind.

685: 2277: 2201: 1282: 773: 572:

from paragraph 4, beginning "Suppose that a boat ...." should be removed - erroneous and speculative.

3289: 2387:

I don't know whether the following explanation will be helpful, but I'm tossing it in for comment.--

1042:

3) If it were possible to go faster downwind by tacking, then why do people bother with spinnakers?

482:

No they don't or there's a revolution in physics happening and it all started in Yachting Monthly.

120: 3416:

No, the issue is that you are refusing to accept multiple citations that disprove your assertion.--

2110: 2068: 2028: 1607: 1495: 1390: 1092: 743: 672: 640: 171:

You don't mean VMG (i.e. velocity made good to the desired point), you mean speed over the ground.

4784: 4765: 4745: 4723: 4704: 4689: 4671: 4614: 4595: 4576: 4556: 4524: 4506: 4491: 4467: 4436: 4407: 4380: 4334: 4252: 4214: 4171: 4135: 4105: 4077: 4044: 4001: 3924: 3890: 3859: 3812: 3765: 3722: 3683: 3654: 3631: 3608: 3590: 3573: 3546: 3528: 3509: 3490: 3471: 3455: 3425: 3383: 3335: 3297: 3251: 3202: 3175: 3144: 3110: 3079: 3052: 3034: 3009: 2983: 2961: 2942: 2925: 2893: 2878: 2848: 2815: 2781: 2759: 2711: 2687: 2672: 2657: 2640: 2623: 2564: 2549:

Neither the mass of the ball nor its KE change as it goes over the sail (Min=Mout and KEin=KEout)

2456: 2442: 2396: 2347: 2321: 2293: 2245: 2217: 2175: 2158: 2136: 2118: 2093: 2072: 2036: 1693: 1630: 1611: 1582: 1568: 1548: 1534: 1510: 1474: 1456: 1441: 1412: 1394: 1369: 1342: 1319: 1298: 1255: 1224: 1197: 1174: 1151: 1130: 1111: 1096: 1031: 1012: 980: 948: 926:" is (still) not representative for a modern yacht (40-50 degrees off true wind is more the case). 897: 874: 840: 821: 802: 785: 764: 731: 706: 676: 660: 644: 629: 602: 491: 465: 443: 422: 396: 373: 338: 322: 297: 258: 239: 218: 143: 128: 112: 4776: 4737: 4715: 4663: 4606: 4587: 4516: 4483: 4459: 4399: 4326: 4244: 4163: 4127: 3993: 3959: 3882: 3804: 3714: 3600: 3565: 3538: 3520: 3482: 3447: 3417: 3136: 3071: 3026: 3001: 2953: 2917: 2885: 2870: 2840: 2799: 2773: 2751: 2703: 2679: 2632: 2615: 2556: 2448: 2434: 2388: 2339: 2313: 2281: 2237: 2205: 2167: 2114: 2085: 2059: 2032: 1685: 1622: 1574: 1560: 1540: 1526: 1502: 1466: 1448: 1433: 1429: 1361: 1334: 1311: 1286: 1247: 1216: 1189: 1166: 1143: 1122: 1103: 1023: 1004: 972: 889: 858: 832: 813: 794: 756: 698: 652: 621: 594: 548: 537:

The major "sources" in the section are to self-published videos demonstrating original research.

457: 388: 330: 250: 104: 2833:, only that the part of the sail on which true wind (i.e. a streamline) acts is moving downwind 2166:

Dear ThinAir, thank you for these. I have incorporated these citations into the main article.--

689: 4542:

wind that is made up mostly of the wind due to its own movement, plus a small reduction and an

3025:

in that direction. Therefore your statement, on which you base your entire argument, is wrong.

2359:

W=F*d, meaning that work W (also called energy) results from force F applied over a distance d.

4285:

This is fundamentally nonsense. I ask you to think again. I have explained why below where at

2055: 1404: 944: 870: 723: 3846:

becomes difficult. The ground is moving, not the boat! A course change requires everything

2019: 1720:

Forward force component (assuming lift coefficient is 1): SIN(Alpha/2)*force of apparent wind

552: 4761: 4700: 4685: 4572: 4552: 4502: 4432: 4376: 4210: 4167: 4101: 4073: 4040: 3997: 3920: 3886: 3855: 3761: 3679: 3650: 3627: 3586: 3569: 3542: 3505: 3467: 3379: 3331: 3247: 3198: 3171: 3140: 3106: 3075: 3048: 3030: 2979: 2938: 2921: 2668: 2653: 2285: 2209: 2051: 2047: 2006: 1437: 1107: 487: 439: 418: 369: 318: 293: 235: 214: 139: 108: 4623:

faster than the true wind (i.e. hullspeed * cosine(angle between course and downwind) : -->

3955: 2264:

A lecture about up-wind-carts & ddwfttw-carts from the Technical University of Denmark

1425: 1499: 4398:

all of that because he thinks that he knows better. It this any way to run Knowledge?--

1723:

Drag force component (drag from the induced headwind): COS(ALPHA)*force of apparent wind

1083:

Sorry, the apparent wind created by the downwind motion of the boat must be vectorially

3135:

I gave a new counter example to your claim based on your clarification. Find it below.

2726:

Simply put, the satisfactory answer is that even if the yacht is going faster - in the

2371: 2064: 1603: 1386: 1088: 668: 636: 119:

baloon argument would be correct). In addition I much agree with your first sentence.

2196:. That is where this quote is from "This confirms that the DDWFTTW condition V/W : --> 1588:

made good (VMG) downwind is zero. This is shown by the value "V0" in the chart below.

1060:

That is, to proceed downwind at the speed of the wind requires essentially no energy.

356:

material there but others such as GL. Anything there has to be properly cited as per

353: 2678:

As shown below, this is not true: they are not wrong, and they do know the basics.--

547:

Possibly a supporting site - but still not up to what I suspect Knowledge needs - ((

4362: 3564:

show that the acceleration can be partially opposed to the apparent wind direction(

3239:(speed over the ground) * cos(angle to the true wind) < (speed of the true wind) 2575: 2043: 1000: 940: 866: 854: 828: 809: 749:

Others (including a physicist) were also skeptical, but then became convinced, see

349: 3047:

We're comparing components of velocities in a particular direction, not speeds (

518:

Certain red flags should prompt editors to examine the sources for a given claim:

409:

I will continue to remove material from the actual article which does not satisfy

3235:(speed over the ground)/cos(angle to the true wind) < (speed of the true wind) 4757: 4696: 4681: 4568: 4548: 4498: 4428: 4372: 4206: 4158: 4097: 4096:

Therefore I disregard any argument that depends upon apparent wind. Good bye!

4069: 4036: 3916: 3851: 3842: 3757: 3675: 3646: 3623: 3582: 3556: 3501: 3463: 3375: 3327: 3243: 3194: 3167: 3102: 2975: 2934: 2664: 2649: 849:

I just removed "Sailing dead downwind faster than the wind" section as there is

483: 435: 414: 410: 365: 361: 357: 314: 289: 269: 231: 210: 135: 46:

If you wish to start a new discussion or revive an old one, please do so on the

2744:

perpendicular to the true wind, can indeed be going more slowly - again in the

521:* surprising or apparently important claims not covered by mainstream sources; 504:

VMG greater than windspeed section does not conform to Knowledge specifications

2772:

such model. In the case above the VMG limit is 1/(1-sin(22.5deg) ~ 1.62 * Vt.

2405:

At that point the boat has kinetic energy, but it's velocity does not change.

2806:) 03:19, 15 June 2010 (UTC) (Corrected which of Newton's laws was applicable 2732:

sense - than the wind in the southerly direction, the portion of the sail,

100:

conditions. I cannot upload files yet, so I put the proposed diagram here:

4497:

I do not have the book. GL said he would send me extracts. Not received.

3850:

to swivel including wind direction, the movement of the ground etc etc.

2447:(updated description of forces' contribution to conservation of momentum 2365: 2719: 313:

point is reached at EXACTLY the same time as the free floating balloon.

996: 533:

If such sources are not available, the material should not be included.

3276:

There is progress here! You (nearly) talk about the same thing: VMG =

2860:

the point at which the sail intersects any given streamline is moving

2146:

1 is achievable with a wheeled vehicle without too much difficulty".

718:

http://www.amazon.com/Symmetry-Sailing-Physics-Yachtsman/dp/1574090003

3561: 2271: 695: 273: 187: 157: 101: 3165:

and I am preparing example(s) to illustrate. Note also that 5 : -->

2197:

1 is achievable with a wheeled vehicle without too much difficulty"

3284:

and cos ≤ 1). The "only" remaining difference now is that VMG : -->

929:

The tables should use a more standard layout (i.e. without border).

4361:

Hi Paul H (this is going to be fun - two Pauls!). If you look at

919:" should use the same terminology (angle -vs- reach/broad reach). 620:

They are supported by numerous other sources, please see below.--

1465:

It is not perpetual motion, please see the explanation above.--

555:)) show an Aussie 18 almost making downwing VMG of windspeed. 4635:

the primary force of the true wind on the boat is against the

2912:

velocity component of the yacht in that particular direction."

2867: 2610:

velocity component of the yacht in that particular direction.

549:

http://www.idniyra.org/articles/IceboatSailingPerformance.html

25: 2546:

Sin = SQRT(2 * KEin / Min ) Sout = SQRT(2 * KEout / Mout)

2374:= 0.5*V^2*m, and it is conserved (unless a force is applied). 528:

Exceptional claims in Knowledge require high-quality sources.

1069:

Talk:Sailing faster than the wind#deleted thought experiment

2018:

Why are we settling on English when this can be settled in

3280:(I guess we all agree it should be * and not / : VMG < 2362:

P=W/t, meaning that power P is work W applied over time t.

2042:

The first problem I have with the diagrams is that force (

553:

http://www.west.net/~lpm/hobie/archives/v1-i3/feature3.htm

2475:

possible even to sail directly upstream on a calm day.

1711:

Angle of apparent wind: 180-(ATAN(SQRT(2)/(SQRT(2)-2*V))

694:. And it is all explained at the bottom of the page at 688:

speed. As stated in the article, the sandyacht data at

2786:

The part of the sail on which the wind acts is indeed,

2368:= V*m, and it is conserved (unless a force is applied). 939:

The language should be clarified/shortened in general.

4482:

to look at Bentwaithe's book, which explains it all.--

2750:

sense - than the true wind in the southerly direction.

2022:? Both Gautier has posted diagrams above, and I have 1278:

perpendicular to boat's motion which never changes.

812:

should be read by anybody interested in this topic.--

103:. If the diagram needs improvement, I can change it. 1048:

5) The cited sources are not sufficiently reliable.

190:

clearly shows a downwind VMG = 1.5 true wind speed.

2262:The following references might be usfull as well: 2014:I think you should be more explicit here in step 2 2795:again, relative to the true wind - see Note below 905:The introduction is too much about speed records. 720:(and it also discusses sailing with a dw vmg: --> 3278:(speed over ground)*cos(angle to the true wind) 1523:The Symmetry of Sailing: The Physics of Sailing 902:What to be cleaned? Here are some suggestions: 3713:please read any elementary manual on flying.-- 514:Exceptional claims require exceptional sources 3223:You misuse the term VMG. By VMG we mean the 2487:in momentum between the wind (billiard ball) 2312:Very helpful, than you. I have added them.-- 2145:confirms that the DDWFTTW condition V/W : --> 691:also show that. As does the iceboat data at 8: 1726:Ratio of drag to forward force: drag/forward 2592:relative to the motion of the second object 1714:Apparent wind speed: SQRT(V^2-V*SQRT(2)+1) 1001:Talk:Sailing#Downwind_faster_than_the_wind 855:Talk:Sailing#Downwind_faster_than_the_wind 810:Talk:Sailing#Downwind_faster_than_the_wind 612:for reasons already stated needs removing. 610:Sailing dead downwind faster than the wind 387:by any citation, in the article itself.-- 2502:medium (true wind, billiard balls, etc) 1554:Sailing at 135 degrees off the true wind 3326:thanks for spotting and fixing my typo 2063:seen so far is similarly unrigorous.-- 44:Do not edit the contents of this page. 744:talk:USA 17 (yacht) § VMG in 1st race 738:Sailing downwind faster than the wind 7: 3881:is the same as in the ground frame. 853:agreement on that in the discussion 559:a simple multiple of apparent wind. 308:what a properly drawn diagram shows 2646:You don't need a diagram, do you? 1003:. I don't know what more to do.-- 24: 2827:saying the sail is moving upwind 1516:Insufficiently reliable citations 917:Sailing perpendicular to the wind 564:Sailing Perpendicular to the Wind 512:Knowledge's recommendations are: 2718: 2574:I had started by saying this is 1668: 1657: 1642: 1525:for Yachtsman, by Ross Garret.-- 865:downwind faster than the wind). 186:This is factually not true. The 29: 3370:of the impossible, one cannot 2194:Drelas general DDWFTTW analysis 2192:twice. Please also include the 1665:that force is sin(sail angle). 2974:It'll take longer than that. 2908:greater in magnitude than the 2863:more slowly than the true wind 2606:greater in magnitude than the 2268:Andrew Bauer's paper from 1969 1704:Alpha: angle of apparent wind 209:showing what you want it to. 1: 2768:downwind (i.e. southward) VMG 2631:diagram that explains this.-- 822:12:16, 18 December 2009 (UTC) 803:12:30, 18 December 2009 (UTC) 786:23:56, 14 December 2009 (UTC) 765:08:39, 10 December 2009 (UTC) 18:Talk:High-performance sailing 1762: 1757: 1752: 1747: 1742: 1737: 1732: 1717:Force of apparent wind: AW^2 1013:10:20, 29 January 2010 (UTC) 981:11:56, 21 January 2010 (UTC) 949:10:43, 20 January 2010 (UTC) 898:13:25, 19 January 2010 (UTC) 875:10:23, 19 January 2010 (UTC) 841:11:26, 13 January 2010 (UTC) 576:Vector diagrams and formulas 2735:on which the true wind acts 4803: 4644:the sail's orientation is 4196:No, I do not. My argument 2789:relative to the true wind, 741: 272:seems to suggest that the 4785:15:26, 10 June 2010 (UTC) 4672:16:26, 17 June 2010 (UTC) 4615:16:26, 17 June 2010 (UTC) 4525:15:28, 10 June 2010 (UTC) 4468:15:56, 10 June 2010 (UTC) 4335:15:53, 10 June 2010 (UTC) 3813:15:49, 10 June 2010 (UTC) 3529:15:35, 10 June 2010 (UTC) 3491:15:35, 10 June 2010 (UTC) 3426:15:35, 10 June 2010 (UTC) 3010:15:35, 10 June 2010 (UTC) 2894:03:19, 15 June 2010 (UTC) 2879:03:19, 15 June 2010 (UTC) 2849:03:19, 15 June 2010 (UTC) 2816:14:55, 17 June 2010 (UTC) 2782:03:19, 15 June 2010 (UTC) 2760:03:19, 15 June 2010 (UTC) 2712:01:24, 16 June 2010 (UTC) 2688:12:46, 15 June 2010 (UTC) 2624:03:19, 15 June 2010 (UTC) 2565:19:17, 15 June 2010 (UTC) 2463:extend this description. 2457:19:27, 15 June 2010 (UTC) 2443:19:17, 15 June 2010 (UTC) 2397:11:32, 15 June 2010 (UTC) 2348:10:43, 15 June 2010 (UTC) 2322:11:42, 15 June 2010 (UTC) 2294:11:25, 15 June 2010 (UTC) 2246:11:42, 15 June 2010 (UTC) 2218:11:11, 15 June 2010 (UTC) 2176:10:21, 15 June 2010 (UTC) 2159:18:41, 14 June 2010 (UTC) 2137:18:35, 14 June 2010 (UTC) 2119:17:44, 14 June 2010 (UTC) 2094:09:52, 15 June 2010 (UTC) 2073:17:15, 14 June 2010 (UTC) 2037:13:31, 14 June 2010 (UTC) 1694:12:41, 15 June 2010 (UTC) 1631:09:12, 14 June 2010 (UTC) 1612:06:43, 14 June 2010 (UTC) 1583:12:27, 11 June 2010 (UTC) 1569:12:27, 11 June 2010 (UTC) 1549:10:16, 15 June 2010 (UTC) 1535:10:16, 15 June 2010 (UTC) 1511:10:16, 15 June 2010 (UTC) 1475:09:03, 14 June 2010 (UTC) 1457:09:54, 14 June 2010 (UTC) 1442:09:12, 14 June 2010 (UTC) 1413:11:46, 14 June 2010 (UTC) 1395:05:20, 14 June 2010 (UTC) 1370:12:35, 14 June 2010 (UTC) 1343:11:57, 14 June 2010 (UTC) 1320:10:56, 14 June 2010 (UTC) 1299:10:00, 14 June 2010 (UTC) 1256:11:53, 14 June 2010 (UTC) 1225:11:42, 14 June 2010 (UTC) 1198:10:56, 14 June 2010 (UTC) 1175:09:27, 14 June 2010 (UTC) 1152:10:39, 14 June 2010 (UTC) 1131:09:03, 14 June 2010 (UTC) 1112:09:12, 14 June 2010 (UTC) 1097:05:20, 14 June 2010 (UTC) 1032:11:22, 11 June 2010 (UTC) 808:Indeed the discussion in 732:11:57, 14 June 2010 (UTC) 707:11:28, 14 June 2010 (UTC) 677:09:43, 14 June 2010 (UTC) 661:09:09, 14 June 2010 (UTC) 645:06:17, 14 June 2010 (UTC) 630:11:41, 14 June 2010 (UTC) 603:11:41, 14 June 2010 (UTC) 259:10:33, 14 June 2010 (UTC) 156:This is exactly what the 4766:08:16, 9 June 2010 (UTC) 4746:07:44, 9 June 2010 (UTC) 4724:08:04, 9 June 2010 (UTC) 4705:20:12, 8 June 2010 (UTC) 4690:20:04, 8 June 2010 (UTC) 4596:08:07, 9 June 2010 (UTC) 4577:19:16, 8 June 2010 (UTC) 4557:18:39, 8 June 2010 (UTC) 4507:08:21, 9 June 2010 (UTC) 4492:07:59, 9 June 2010 (UTC) 4437:19:34, 8 June 2010 (UTC) 4408:07:59, 9 June 2010 (UTC) 4381:18:15, 8 June 2010 (UTC) 4253:08:13, 9 June 2010 (UTC) 4215:07:38, 9 June 2010 (UTC) 4172:05:55, 9 June 2010 (UTC) 4136:07:59, 9 June 2010 (UTC) 4106:19:34, 8 June 2010 (UTC) 4078:19:34, 8 June 2010 (UTC) 4045:07:44, 9 June 2010 (UTC) 4002:08:14, 9 June 2010 (UTC) 3925:07:44, 9 June 2010 (UTC) 3891:05:55, 9 June 2010 (UTC) 3860:19:34, 8 June 2010 (UTC) 3766:08:26, 9 June 2010 (UTC) 3723:07:59, 9 June 2010 (UTC) 3684:19:51, 8 June 2010 (UTC) 3655:16:37, 8 June 2010 (UTC) 3632:08:29, 9 June 2010 (UTC) 3609:07:59, 9 June 2010 (UTC) 3591:16:14, 8 June 2010 (UTC) 3574:15:52, 8 June 2010 (UTC) 3547:11:14, 9 June 2010 (UTC) 3510:07:56, 9 June 2010 (UTC) 3472:07:54, 9 June 2010 (UTC) 3456:07:49, 9 June 2010 (UTC) 3384:07:47, 9 June 2010 (UTC) 3336:07:51, 9 June 2010 (UTC) 3298:07:39, 9 June 2010 (UTC) 3252:07:18, 9 June 2010 (UTC) 3203:06:45, 9 June 2010 (UTC) 3176:06:26, 9 June 2010 (UTC) 3145:11:14, 9 June 2010 (UTC) 3111:08:07, 9 June 2010 (UTC) 3080:07:57, 9 June 2010 (UTC) 3053:06:26, 9 June 2010 (UTC) 3035:22:22, 8 June 2010 (UTC) 2984:08:07, 9 June 2010 (UTC) 2962:07:47, 9 June 2010 (UTC) 2943:16:09, 8 June 2010 (UTC) 2926:15:52, 8 June 2010 (UTC) 2673:14:57, 8 June 2010 (UTC) 2663:if they ever knew them. 2658:14:52, 8 June 2010 (UTC) 2641:07:44, 9 June 2010 (UTC) 2514:in momentum between the 1707:AW: Apparent wind speed 913:Sailing on a broad reach 570:Sailing on a Broad Reach 492:15:43, 8 June 2010 (UTC) 466:09:11, 8 June 2010 (UTC) 444:08:58, 8 June 2010 (UTC) 423:08:58, 8 June 2010 (UTC) 397:09:11, 8 June 2010 (UTC) 374:08:58, 8 June 2010 (UTC) 339:08:46, 8 June 2010 (UTC) 323:08:33, 8 June 2010 (UTC) 298:15:13, 8 June 2010 (UTC) 240:08:18, 8 June 2010 (UTC) 219:15:13, 8 June 2010 (UTC) 144:00:49, 8 June 2010 (UTC) 129:23:38, 7 June 2010 (UTC) 113:22:13, 7 June 2010 (UTC) 2551:in this reference frame 936:" should be cleaned up. 274:proposed vector diagram 188:proposed vector diagram 158:proposed vector diagram 3286:speed of the true wind 1749:Force of apparent wind 1142:component of the wind. 1052:Conservation of energy 2952:take me a few days.-- 2830:relative to the water 2190:Drelas power analysis 1680:Force and speed table 1075:No force on the sails 42:of past discussions. 4368:on every square inch 3481:was downwind VMG.-- 2839:than the true wind. 2383:Billiard ball model 2236:Fixed, thank you.-- 1744:Apparent wind speed 3960:Lift-to-drag_ratio 3237: 2911: 2907: 2609: 2605: 2570:Physical mechanics 2188:You have included 2060:lift-to-drag ratio 1764:Ratio drag/forward 1698:The formulas are: 1496:2010 America's Cup 1481:Why use spinnakers 1430:Lift-to-drag_ratio 4650:to the true wind, 4647:not perpendicular 3374:the impossible. 3282:speed over ground 3233: 2909: 2905: 2607: 2603: 2539:KE = M * S^2 / 2 2430:Billiard balls II 2297: 2280:comment added by 2221: 2204:comment added by 2056:lift-induced drag 1995: 1994: 1754:Forward component 1302: 1285:comment added by 827:In partcular the 776:comment added by 92: 91: 54: 53: 48:current talk page 4794: 4458:out of order".-- 2722: 2296: 2274: 2272:DDWFTTW-vechicle 2220: 2198: 2052:drag coefficient 2048:lift coefficient 2007:International DN 1730: 1729: 1672: 1661: 1646: 1301: 1279: 1018:Common fallacies 788: 551:)). Polars at (( 73: 56: 55: 33: 32: 26: 4802: 4801: 4797: 4796: 4795: 4793: 4792: 4791: 4734: 4535: 3956:Angle_of_attack 3841:Maybe the term 3562:vector diagrams 3554: 2901: 2792:moving upwind ( 2696: 2572: 2547: 2540: 2534: 2505:does not matter 2432: 2385: 2336: 2275: 2199: 2024:posted diagrams 2016: 1682: 1556: 1518: 1492: 1483: 1426:Angle_of_attack 1280: 1077: 1054: 1020: 997:sailing article 934:Further Reading 771: 746: 740: 582:Speed made good 525: 522: 506: 310: 97: 69: 30: 22: 21: 20: 12: 11: 5: 4800: 4798: 4790: 4789: 4788: 4787: 4769: 4768: 4758:Paul Beardsell 4733: 4730: 4729: 4728: 4727: 4726: 4708: 4707: 4697:Paul Beardsell 4677: 4676: 4675: 4674: 4656: 4655: 4654: 4653: 4652: 4651: 4642: 4628: 4627: 4626: 4625: 4601: 4600: 4599: 4598: 4580: 4579: 4569:Paul Beardsell 4534: 4531: 4530: 4529: 4528: 4527: 4510: 4509: 4499:Paul Beardsell 4479: 4478: 4477: 4476: 4475: 4474: 4473: 4472: 4471: 4470: 4446: 4445: 4444: 4443: 4442: 4441: 4440: 4439: 4429:Paul Beardsell 4417: 4416: 4415: 4414: 4413: 4412: 4411: 4410: 4388: 4387: 4386: 4385: 4384: 4383: 4354: 4353: 4352: 4351: 4350: 4349: 4348: 4347: 4346: 4345: 4344: 4343: 4342: 4341: 4340: 4339: 4338: 4337: 4305: 4304: 4303: 4302: 4301: 4300: 4299: 4298: 4297: 4296: 4295: 4294: 4293: 4292: 4291: 4290: 4268: 4267: 4266: 4265: 4264: 4263: 4262: 4261: 4260: 4259: 4258: 4257: 4256: 4255: 4228: 4227: 4226: 4225: 4224: 4223: 4222: 4221: 4220: 4219: 4218: 4217: 4207:Paul Beardsell 4183: 4182: 4181: 4180: 4179: 4178: 4177: 4176: 4175: 4174: 4159:Paul Beardsell 4147: 4146: 4145: 4144: 4143: 4142: 4141: 4140: 4139: 4138: 4115: 4114: 4113: 4112: 4111: 4110: 4109: 4108: 4098:Paul Beardsell 4087: 4086: 4085: 4084: 4083: 4082: 4081: 4080: 4070:Paul Beardsell 4058: 4057: 4056: 4055: 4054: 4053: 4052: 4051: 4050: 4049: 4048: 4047: 4037:Paul Beardsell 4017: 4016: 4015: 4014: 4013: 4012: 4011: 4010: 4009: 4008: 4007: 4006: 4005: 4004: 3976: 3975: 3974: 3973: 3972: 3971: 3970: 3969: 3968: 3967: 3966: 3965: 3964: 3963: 3938: 3937: 3936: 3935: 3934: 3933: 3932: 3931: 3930: 3929: 3928: 3927: 3917:Paul Beardsell 3902: 3901: 3900: 3899: 3898: 3897: 3896: 3895: 3894: 3893: 3869: 3868: 3867: 3866: 3865: 3864: 3863: 3862: 3852:Paul Beardsell 3832: 3831: 3830: 3829: 3828: 3827: 3826: 3825: 3824: 3823: 3822: 3821: 3820: 3819: 3818: 3817: 3816: 3815: 3783: 3782: 3781: 3780: 3779: 3778: 3777: 3776: 3775: 3774: 3773: 3772: 3771: 3770: 3769: 3768: 3758:Paul Beardsell 3738: 3737: 3736: 3735: 3734: 3733: 3732: 3731: 3730: 3729: 3728: 3727: 3726: 3725: 3697: 3696: 3695: 3694: 3693: 3692: 3691: 3690: 3689: 3688: 3687: 3686: 3676:Paul Beardsell 3660: 3659: 3658: 3657: 3639: 3638: 3637: 3636: 3635: 3634: 3624:Paul Beardsell 3614: 3613: 3612: 3611: 3594: 3593: 3583:Paul Beardsell 3557:Paul Beardsell 3553: 3550: 3534: 3533: 3532: 3531: 3513: 3512: 3502:Paul Beardsell 3496: 3495: 3494: 3493: 3475: 3474: 3464:Paul Beardsell 3443: 3442: 3441: 3440: 3439: 3438: 3437: 3436: 3435: 3434: 3433: 3432: 3431: 3430: 3429: 3428: 3399: 3398: 3397: 3396: 3395: 3394: 3393: 3392: 3391: 3390: 3389: 3388: 3387: 3386: 3376:Paul Beardsell 3351: 3350: 3349: 3348: 3347: 3346: 3345: 3344: 3343: 3342: 3341: 3340: 3339: 3338: 3328:Paul Beardsell 3311: 3310: 3309: 3308: 3307: 3306: 3305: 3304: 3303: 3302: 3301: 3300: 3263: 3262: 3261: 3260: 3259: 3258: 3257: 3256: 3255: 3254: 3244:Paul Beardsell 3212: 3211: 3210: 3209: 3208: 3207: 3206: 3205: 3183: 3182: 3181: 3180: 3179: 3178: 3168:Paul Beardsell 3158: 3157: 3156: 3155: 3154: 3153: 3152: 3151: 3150: 3149: 3148: 3147: 3122: 3121: 3120: 3119: 3118: 3117: 3116: 3115: 3114: 3113: 3103:Paul Beardsell 3089: 3088: 3087: 3086: 3085: 3084: 3083: 3082: 3060: 3059: 3058: 3057: 3056: 3055: 3040: 3039: 3038: 3037: 3019: 3018: 3017: 3016: 3015: 3014: 3013: 3012: 2991: 2990: 2989: 2988: 2987: 2986: 2976:Paul Beardsell 2967: 2966: 2965: 2964: 2946: 2945: 2935:Paul Beardsell 2900: 2897: 2725: 2717: 2695: 2694:Criticism zero 2692: 2691: 2690: 2665:Paul Beardsell 2650:Paul Beardsell 2644: 2643: 2571: 2568: 2545: 2538: 2532: 2431: 2428: 2426: 2384: 2381: 2376: 2375: 2372:Kinetic energy 2369: 2363: 2360: 2357: 2354: 2335: 2334:Basic concepts 2332: 2331: 2330: 2329: 2328: 2327: 2326: 2325: 2324: 2303: 2302: 2301: 2300: 2299: 2298: 2255: 2254: 2253: 2252: 2251: 2250: 2249: 2248: 2227: 2226: 2225: 2224: 2223: 2222: 2181: 2180: 2179: 2178: 2151:ThinAirDesigns 2142: 2141: 2140: 2139: 2129:ThinAirDesigns 2121: 2101: 2100: 2099: 2098: 2097: 2096: 2076: 2075: 2015: 2012: 1993: 1992: 1989: 1988: 1985: 1982: 1979: 1976: 1973: 1970: 1966: 1965: 1962: 1959: 1956: 1953: 1950: 1947: 1943: 1942: 1939: 1936: 1933: 1930: 1927: 1924: 1920: 1919: 1916: 1913: 1910: 1907: 1904: 1901: 1897: 1896: 1893: 1890: 1887: 1884: 1881: 1878: 1874: 1873: 1870: 1867: 1864: 1861: 1858: 1855: 1851: 1850: 1848: 1846: 1843: 1840: 1837: 1834: 1830: 1829: 1827: 1825: 1822: 1819: 1816: 1813: 1809: 1808: 1806: 1804: 1801: 1798: 1795: 1792: 1788: 1787: 1785: 1783: 1780: 1777: 1774: 1771: 1767: 1766: 1761: 1759:Drag component 1756: 1751: 1746: 1741: 1736: 1728: 1727: 1724: 1721: 1718: 1715: 1712: 1701:V: boat speed 1681: 1678: 1636: 1635: 1634: 1633: 1615: 1614: 1555: 1552: 1517: 1514: 1491: 1490:Published data 1488: 1482: 1479: 1478: 1477: 1462: 1461: 1460: 1459: 1444: 1418: 1417: 1416: 1415: 1400: 1379: 1378: 1377: 1376: 1375: 1374: 1373: 1372: 1350: 1349: 1348: 1347: 1346: 1345: 1325: 1324: 1323: 1322: 1304: 1303: 1267: 1266: 1265: 1264: 1263: 1262: 1261: 1260: 1259: 1258: 1234: 1233: 1232: 1231: 1230: 1229: 1228: 1227: 1205: 1204: 1203: 1202: 1201: 1200: 1180: 1179: 1178: 1177: 1159: 1158: 1157: 1156: 1155: 1154: 1134: 1133: 1117: 1116: 1115: 1114: 1076: 1073: 1053: 1050: 1019: 1016: 992: 991: 990: 989: 988: 987: 986: 985: 984: 983: 960: 959: 958: 957: 956: 955: 954: 953: 952: 951: 937: 930: 927: 920: 911:The headings " 909: 906: 880: 879: 878: 877: 844: 843: 806: 805: 778:202.180.87.174 739: 736: 735: 734: 712: 711: 710: 709: 664: 663: 633: 632: 614: 613: 606: 605: 587: 586: 579: 573: 567: 523: 520: 505: 502: 501: 500: 499: 498: 497: 496: 495: 494: 484:Paul Beardsell 473: 472: 471: 470: 469: 468: 449: 448: 447: 446: 436:Paul Beardsell 428: 427: 426: 425: 415:Paul Beardsell 404: 403: 402: 401: 400: 399: 379: 378: 377: 376: 366:Paul Beardsell 342: 341: 315:Paul Beardsell 309: 306: 305: 304: 303: 302: 301: 300: 290:Paul Beardsell 280: 279: 278: 277: 270:Paul Beardsell 264: 263: 262: 261: 243: 242: 232:Paul Beardsell 226: 225: 224: 223: 222: 221: 211:Paul Beardsell 198: 197: 196: 195: 194: 193: 192: 191: 177: 176: 175: 174: 173: 172: 164: 163: 162: 161: 151: 150: 132: 131: 96: 95:Simple Diagram 93: 90: 89: 84: 79: 74: 67: 62: 52: 51: 34: 23: 15: 14: 13: 10: 9: 6: 4: 3: 2: 4799: 4786: 4782: 4778: 4777:Gautier lebon 4773: 4772: 4771: 4770: 4767: 4763: 4759: 4755: 4754:apparent wind 4750: 4749: 4748: 4747: 4743: 4739: 4738:Gautier lebon 4732:criticism 4 = 4731: 4725: 4721: 4717: 4716:Gautier lebon 4712: 4711: 4710: 4709: 4706: 4702: 4698: 4694: 4693: 4692: 4691: 4687: 4683: 4673: 4669: 4665: 4664:Sirclicksalot 4660: 4659: 4658: 4657: 4649: 4648: 4643: 4640: 4639: 4634: 4633: 4632: 4631: 4630: 4629: 4622: 4621: 4616: 4612: 4608: 4607:Sirclicksalot 4605: 4604: 4603: 4602: 4597: 4593: 4589: 4588:Gautier lebon 4584: 4583: 4582: 4581: 4578: 4574: 4570: 4565: 4561: 4560: 4559: 4558: 4554: 4550: 4545: 4541: 4532: 4526: 4522: 4518: 4517:Gautier lebon 4514: 4513: 4512: 4511: 4508: 4504: 4500: 4496: 4495: 4494: 4493: 4489: 4485: 4484:Gautier lebon 4469: 4465: 4461: 4460:Gautier lebon 4456: 4455: 4454: 4453: 4452: 4451: 4450: 4449: 4448: 4447: 4438: 4434: 4430: 4425: 4424: 4423: 4422: 4421: 4420: 4419: 4418: 4409: 4405: 4401: 4400:Gautier lebon 4396: 4395: 4394: 4393: 4392: 4391: 4390: 4389: 4382: 4378: 4374: 4369: 4364: 4360: 4359: 4358: 4357: 4356: 4355: 4336: 4332: 4328: 4327:Gautier lebon 4323: 4322: 4321: 4320: 4319: 4318: 4317: 4316: 4315: 4314: 4313: 4312: 4311: 4310: 4309: 4308: 4307: 4306: 4288: 4284: 4283: 4282: 4281: 4280: 4279: 4278: 4277: 4276: 4275: 4274: 4273: 4272: 4271: 4270: 4269: 4254: 4250: 4246: 4245:Gautier lebon 4242: 4241: 4240: 4239: 4238: 4237: 4236: 4235: 4234: 4233: 4232: 4231: 4230: 4229: 4216: 4212: 4208: 4204: 4199: 4195: 4194: 4193: 4192: 4191: 4190: 4189: 4188: 4187: 4186: 4185: 4184: 4173: 4169: 4165: 4160: 4157: 4156: 4155: 4154: 4153: 4152: 4151: 4150: 4149: 4148: 4137: 4133: 4129: 4128:Gautier lebon 4125: 4124: 4123: 4122: 4121: 4120: 4119: 4118: 4117: 4116: 4107: 4103: 4099: 4095: 4094: 4093: 4092: 4091: 4090: 4089: 4088: 4079: 4075: 4071: 4066: 4065: 4064: 4063: 4062: 4061: 4060: 4059: 4046: 4042: 4038: 4034: 4029: 4028: 4027: 4026: 4025: 4024: 4023: 4022: 4021: 4020: 4019: 4018: 4003: 3999: 3995: 3990: 3989: 3988: 3987: 3986: 3985: 3984: 3983: 3982: 3981: 3980: 3979: 3978: 3977: 3961: 3957: 3952: 3951: 3950: 3949: 3948: 3947: 3946: 3945: 3944: 3943: 3942: 3941: 3940: 3939: 3926: 3922: 3918: 3914: 3913: 3912: 3911: 3910: 3909: 3908: 3907: 3906: 3905: 3904: 3903: 3892: 3888: 3884: 3879: 3878: 3877: 3876: 3875: 3874: 3873: 3872: 3871: 3870: 3861: 3857: 3853: 3849: 3844: 3840: 3839: 3838: 3837: 3836: 3835: 3834: 3833: 3814: 3810: 3806: 3805:Gautier lebon 3801: 3800: 3799: 3798: 3797: 3796: 3795: 3794: 3793: 3792: 3791: 3790: 3789: 3788: 3787: 3786: 3785: 3784: 3767: 3763: 3759: 3754: 3753: 3752: 3751: 3750: 3749: 3748: 3747: 3746: 3745: 3744: 3743: 3742: 3741: 3740: 3739: 3724: 3720: 3716: 3715:Gautier lebon 3711: 3710: 3709: 3708: 3707: 3706: 3705: 3704: 3703: 3702: 3701: 3700: 3699: 3698: 3685: 3681: 3677: 3672: 3671: 3670: 3669: 3668: 3667: 3666: 3665: 3664: 3663: 3662: 3661: 3656: 3652: 3648: 3643: 3642: 3641: 3640: 3633: 3629: 3625: 3620: 3619: 3618: 3617: 3616: 3615: 3610: 3606: 3602: 3601:Gautier lebon 3598: 3597: 3596: 3595: 3592: 3588: 3584: 3579: 3578: 3577: 3575: 3571: 3567: 3563: 3558: 3551: 3549: 3548: 3544: 3540: 3530: 3526: 3522: 3521:Gautier lebon 3517: 3516: 3515: 3514: 3511: 3507: 3503: 3498: 3497: 3492: 3488: 3484: 3483:Gautier lebon 3479: 3478: 3477: 3476: 3473: 3469: 3465: 3460: 3459: 3458: 3457: 3453: 3449: 3448:Gautier lebon 3427: 3423: 3419: 3418:Gautier lebon 3415: 3414: 3413: 3412: 3411: 3410: 3409: 3408: 3407: 3406: 3405: 3404: 3403: 3402: 3401: 3400: 3385: 3381: 3377: 3373: 3369: 3365: 3364: 3363: 3362: 3361: 3360: 3359: 3358: 3357: 3356: 3355: 3354: 3353: 3352: 3337: 3333: 3329: 3325: 3324: 3323: 3322: 3321: 3320: 3319: 3318: 3317: 3316: 3315: 3314: 3313: 3312: 3299: 3295: 3291: 3287: 3283: 3279: 3275: 3274: 3273: 3272: 3271: 3270: 3269: 3268: 3267: 3266: 3265: 3264: 3253: 3249: 3245: 3240: 3236: 3230: 3226: 3222: 3221: 3220: 3219: 3218: 3217: 3216: 3215: 3214: 3213: 3204: 3200: 3196: 3191: 3190: 3189: 3188: 3187: 3186: 3185: 3184: 3177: 3173: 3169: 3164: 3163: 3162: 3161: 3160: 3159: 3146: 3142: 3138: 3134: 3133: 3132: 3131: 3130: 3129: 3128: 3127: 3126: 3125: 3124: 3123: 3112: 3108: 3104: 3099: 3098: 3097: 3096: 3095: 3094: 3093: 3092: 3091: 3090: 3081: 3077: 3073: 3068: 3067: 3066: 3065: 3064: 3063: 3062: 3061: 3054: 3050: 3046: 3045: 3044: 3043: 3042: 3041: 3036: 3032: 3028: 3023: 3022: 3021: 3020: 3011: 3007: 3003: 3002:Gautier lebon 2999: 2998: 2997: 2996: 2995: 2994: 2993: 2992: 2985: 2981: 2977: 2973: 2972: 2971: 2970: 2969: 2968: 2963: 2959: 2955: 2954:Gautier lebon 2950: 2949: 2948: 2947: 2944: 2940: 2936: 2931: 2930: 2929: 2927: 2923: 2919: 2914: 2913: 2898: 2896: 2895: 2891: 2887: 2886:Sirclicksalot 2881: 2880: 2876: 2872: 2871:Sirclicksalot 2869: 2865: 2864: 2858: 2857: 2851: 2850: 2846: 2842: 2841:Sirclicksalot 2838: 2837: 2832: 2831: 2826: 2825: 2819: 2817: 2813: 2809: 2805: 2801: 2800:Sirclicksalot 2797: 2796: 2791: 2790: 2784: 2783: 2779: 2775: 2774:Sirclicksalot 2770: 2769: 2762: 2761: 2757: 2753: 2752:Sirclicksalot 2749: 2748: 2743: 2742: 2738:and which is 2737: 2736: 2731: 2730: 2723: 2721: 2715: 2713: 2709: 2705: 2704:Sirclicksalot 2701: 2693: 2689: 2685: 2681: 2680:Gautier lebon 2677: 2676: 2675: 2674: 2670: 2666: 2660: 2659: 2655: 2651: 2647: 2642: 2638: 2634: 2633:Gautier lebon 2629: 2628: 2627: 2625: 2621: 2617: 2616:Sirclicksalot 2611: 2600: 2597: 2593: 2588: 2585: 2581: 2577: 2569: 2567: 2566: 2562: 2558: 2557:Sirclicksalot 2554: 2552: 2544: 2537: 2531: 2528: 2524: 2521: 2517: 2513: 2512: 2507: 2506: 2501: 2500: 2494: 2490: 2486: 2480: 2476: 2474: 2473:theoretically 2470: 2464: 2460: 2458: 2454: 2450: 2449:Sirclicksalot 2445: 2444: 2440: 2436: 2435:Sirclicksalot 2429: 2427: 2424: 2421: 2418: 2414: 2410: 2406: 2403: 2399: 2398: 2394: 2390: 2389:Gautier lebon 2382: 2380: 2373: 2370: 2367: 2364: 2361: 2358: 2355: 2352: 2351: 2350: 2349: 2345: 2341: 2340:Gautier lebon 2333: 2323: 2319: 2315: 2314:Gautier lebon 2311: 2310: 2309: 2308: 2307: 2306: 2305: 2304: 2295: 2291: 2287: 2283: 2279: 2273: 2269: 2265: 2261: 2260: 2259: 2258: 2257: 2256: 2247: 2243: 2239: 2238:Gautier lebon 2235: 2234: 2233: 2232: 2231: 2230: 2229: 2228: 2219: 2215: 2211: 2207: 2203: 2195: 2191: 2187: 2186: 2185: 2184: 2183: 2182: 2177: 2173: 2169: 2168:Gautier lebon 2165: 2164: 2163: 2162: 2161: 2160: 2156: 2152: 2149: 2138: 2134: 2130: 2127: 2122: 2120: 2116: 2112: 2105: 2104: 2103: 2102: 2095: 2091: 2087: 2086:Gautier lebon 2082: 2081: 2080: 2079: 2078: 2077: 2074: 2070: 2066: 2061: 2057: 2053: 2049: 2045: 2041: 2040: 2039: 2038: 2034: 2030: 2025: 2021: 2013: 2011: 2008: 2003: 1999: 1991: 1990: 1986: 1983: 1980: 1977: 1974: 1971: 1968: 1967: 1963: 1960: 1957: 1954: 1951: 1948: 1945: 1944: 1940: 1937: 1934: 1931: 1928: 1925: 1922: 1921: 1917: 1914: 1911: 1908: 1905: 1902: 1899: 1898: 1894: 1891: 1888: 1885: 1882: 1879: 1876: 1875: 1871: 1868: 1865: 1862: 1859: 1856: 1853: 1852: 1849: 1847: 1844: 1841: 1838: 1835: 1832: 1831: 1828: 1826: 1823: 1820: 1817: 1814: 1811: 1810: 1807: 1805: 1802: 1799: 1796: 1793: 1790: 1789: 1786: 1784: 1781: 1778: 1775: 1772: 1769: 1768: 1765: 1760: 1755: 1750: 1745: 1740: 1735: 1731: 1725: 1722: 1719: 1716: 1713: 1710: 1709: 1708: 1705: 1702: 1699: 1696: 1695: 1691: 1687: 1686:Gautier lebon 1679: 1677: 1673: 1671: 1666: 1662: 1660: 1655: 1651: 1647: 1645: 1640: 1632: 1628: 1624: 1623:Gautier lebon 1619: 1618: 1617: 1616: 1613: 1609: 1605: 1600: 1599: 1598: 1597: 1593: 1589: 1585: 1584: 1580: 1576: 1575:Gautier lebon 1571: 1570: 1566: 1562: 1561:Gautier lebon 1553: 1551: 1550: 1546: 1542: 1541:Gautier lebon 1537: 1536: 1532: 1528: 1527:Gautier lebon 1524: 1515: 1513: 1512: 1508: 1504: 1503:Gautier lebon 1500: 1497: 1489: 1487: 1480: 1476: 1472: 1468: 1467:Gautier lebon 1464: 1463: 1458: 1454: 1450: 1449:Sirclicksalot 1445: 1443: 1439: 1435: 1431: 1427: 1422: 1421: 1420: 1419: 1414: 1410: 1406: 1401: 1398: 1397: 1396: 1392: 1388: 1384: 1383: 1382: 1371: 1367: 1363: 1362:Gautier lebon 1358: 1357: 1356: 1355: 1354: 1353: 1352: 1351: 1344: 1340: 1336: 1335:Sirclicksalot 1331: 1330: 1329: 1328: 1327: 1326: 1321: 1317: 1313: 1312:Gautier lebon 1308: 1307: 1306: 1305: 1300: 1296: 1292: 1288: 1287:Sirclicksalot 1284: 1276: 1275: 1274: 1270: 1257: 1253: 1249: 1248:Gautier lebon 1244: 1243: 1242: 1241: 1240: 1239: 1238: 1237: 1236: 1235: 1226: 1222: 1218: 1217:Sirclicksalot 1213: 1212: 1211: 1210: 1209: 1208: 1207: 1206: 1199: 1195: 1191: 1190:Gautier lebon 1186: 1185: 1184: 1183: 1182: 1181: 1176: 1172: 1168: 1167:Sirclicksalot 1163: 1162: 1161: 1160: 1153: 1149: 1145: 1144:Sirclicksalot 1140: 1139: 1138: 1137: 1136: 1135: 1132: 1128: 1124: 1123:Gautier lebon 1119: 1118: 1113: 1109: 1105: 1100: 1099: 1098: 1094: 1090: 1086: 1082: 1081: 1080: 1074: 1072: 1070: 1065: 1061: 1058: 1051: 1049: 1046: 1043: 1040: 1037: 1034: 1033: 1029: 1025: 1024:Gautier lebon 1017: 1015: 1014: 1010: 1006: 1005:Gautier lebon 1002: 998: 982: 978: 974: 973:Gautier lebon 970: 969: 968: 967: 966: 965: 964: 963: 962: 961: 950: 946: 942: 938: 935: 931: 928: 925: 921: 918: 914: 910: 907: 904: 903: 901: 900: 899: 895: 891: 890:Gautier lebon 886: 885: 884: 883: 882: 881: 876: 872: 868: 864: 860: 859:Gautier lebon 856: 852: 848: 847: 846: 845: 842: 838: 834: 833:Gautier lebon 830: 826: 825: 824: 823: 819: 815: 814:Gautier lebon 811: 804: 800: 796: 795:Gautier lebon 791: 790: 789: 787: 783: 779: 775: 767: 766: 762: 758: 757:Gautier lebon 754: 751: 745: 737: 733: 729: 725: 719: 714: 713: 708: 704: 700: 699:Gautier lebon 696: 693: 690: 686: 683: 682: 681: 680: 679: 678: 674: 670: 662: 658: 654: 653:Gautier lebon 649: 648: 647: 646: 642: 638: 631: 627: 623: 622:Gautier lebon 619: 618: 617: 611: 608: 607: 604: 600: 596: 595:Gautier lebon 592: 589: 588: 583: 580: 577: 574: 571: 568: 565: 562: 561: 560: 556: 554: 550: 545: 542: 538: 535: 534: 530: 529: 519: 516: 515: 510: 503: 493: 489: 485: 481: 480: 479: 478: 477: 476: 475: 474: 467: 463: 459: 458:Gautier lebon 455: 454: 453: 452: 451: 450: 445: 441: 437: 432: 431: 430: 429: 424: 420: 416: 412: 408: 407: 406: 405: 398: 394: 390: 389:Gautier lebon 385: 384: 383: 382: 381: 380: 375: 371: 367: 363: 359: 355: 351: 346: 345: 344: 343: 340: 336: 332: 331:Gautier lebon 327: 326: 325: 324: 320: 316: 307: 299: 295: 291: 286: 285: 284: 283: 282: 281: 275: 271: 268: 267: 266: 265: 260: 256: 252: 251:Sirclicksalot 247: 246: 245: 244: 241: 237: 233: 228: 227: 220: 216: 212: 208: 204: 203: 202: 201: 200: 199: 189: 185: 184: 183: 182: 181: 180: 179: 178: 170: 169: 168: 167: 166: 165: 159: 155: 154: 153: 152: 148: 147: 146: 145: 141: 137: 130: 126: 122: 117: 116: 115: 114: 110: 106: 102: 94: 88: 85: 83: 80: 78: 75: 72: 68: 66: 63: 61: 58: 57: 49: 45: 41: 40: 35: 28: 27: 19: 4753: 4735: 4678: 4646: 4645: 4637: 4636: 4619: 4618: 4543: 4539: 4536: 4480: 4367: 4363:Lift (force) 4286: 4202: 4197: 4032: 3847: 3555: 3535: 3444: 3371: 3367: 3285: 3281: 3277: 3238: 3234: 3228: 3224: 2915: 2903: 2902: 2882: 2862: 2861: 2855: 2854: 2852: 2835: 2834: 2829: 2828: 2823: 2822: 2821:Note: I am 2820: 2808:74.79.24.116 2794: 2793: 2788: 2787: 2785: 2767: 2766: 2763: 2746: 2745: 2740: 2739: 2734: 2733: 2728: 2727: 2724: 2716: 2699: 2697: 2661: 2648: 2645: 2612: 2601: 2595: 2591: 2589: 2583: 2579: 2573: 2555: 2550: 2548: 2541: 2535: 2529: 2525: 2519: 2515: 2510: 2509: 2504: 2503: 2498: 2497: 2492: 2488: 2484: 2481: 2477: 2472: 2468: 2465: 2461: 2446: 2433: 2425: 2422: 2419: 2415: 2411: 2407: 2404: 2400: 2386: 2377: 2337: 2143: 2017: 2005:The popular 2004: 2000: 1996: 1763: 1758: 1753: 1748: 1743: 1738: 1733: 1706: 1703: 1700: 1697: 1683: 1674: 1667: 1663: 1656: 1652: 1648: 1641: 1637: 1595: 1594: 1590: 1586: 1572: 1557: 1538: 1522: 1519: 1493: 1484: 1405:Waleswatcher 1380: 1271: 1268: 1084: 1078: 1066: 1062: 1059: 1055: 1047: 1044: 1041: 1038: 1035: 1021: 993: 933: 923: 916: 912: 862: 850: 807: 768: 747: 724:Waleswatcher 665: 634: 615: 609: 581: 575: 569: 563: 557: 546: 543: 539: 536: 532: 531: 527: 526: 517: 513: 511: 507: 311: 206: 133: 98: 70: 43: 37: 4533:criticism 3 4427:of this.) 4287:criticism 4 4033:equivalence 3843:red herring 3552:criticism 2 3193:speed????-- 2899:criticism 1 2836:more slowly 2580:other stuff 2508:, only the 2276:—Preceding 2200:—Preceding 1281:—Preceding 772:—Preceding 36:This is an 4564:Newton One 2533:M = m * V 2270:about his 1734:Boat speed 1085:subtracted 742:See also: 207:supposedly 3992:approach. 3290:MrBeanBob 2065:QuietJohn 1604:QuietJohn 1387:QuietJohn 1089:QuietJohn 669:QuietJohn 637:QuietJohn 121:MrBeanBob 87:Archive 6 82:Archive 5 77:Archive 4 71:Archive 3 65:Archive 2 60:Archive 1 4620:downwind 4540:apparent 4203:apparent 3803:speed.-- 3519:wind).-- 3225:velocity 2856:However, 2520:outgoing 2516:incoming 2499:incoming 2366:Momentum 2290:contribs 2278:unsigned 2214:contribs 2202:unsigned 2111:Generacy 2029:Generacy 1295:contribs 1283:unsigned 1246:clear.-- 774:unsigned 329:later.-- 4198:depends 941:Prillen 915:" and " 867:Prillen 829:summary 39:archive 4682:Nigelj 4549:Nigelj 4373:Nigelj 4164:Eyytee 3994:Eyytee 3883:Eyytee 3566:Eyytee 3539:Eyytee 3137:Eyytee 3072:Eyytee 3027:Eyytee 2918:Eyytee 2906:faster 2604:faster 2511:change 2489:before 2485:change 2469:ground 2282:Eyytee 2206:Eyytee 1987:12.10 1434:Eyytee 1104:Eyytee 354:WP:SYN 105:Eyytee 4641:, and 4544:angle 3229:speed 2910:speed 2608:speed 2582:must 2576:WP:OR 2493:after 2027:that. 1984:18.68 1978:18.93 1964:9.26 1961:11.09 1955:11.34 1941:6.41 1918:3.54 1895:1.85 1872:0.69 1739:Alpha 350:WP:OR 16:< 4781:talk 4762:talk 4742:talk 4720:talk 4701:talk 4686:talk 4668:talk 4638:sail 4611:talk 4592:talk 4573:talk 4553:talk 4521:talk 4503:talk 4488:talk 4464:talk 4433:talk 4404:talk 4377:talk 4331:talk 4249:talk 4211:talk 4168:talk 4132:talk 4102:talk 4074:talk 4041:talk 3998:talk 3958:and 3921:talk 3887:talk 3856:talk 3848:else 3809:talk 3762:talk 3719:talk 3680:talk 3651:talk 3647:Paul 3628:talk 3605:talk 3587:talk 3570:talk 3543:talk 3525:talk 3506:talk 3487:talk 3468:talk 3452:talk 3422:talk 3380:talk 3368:talk 3332:talk 3294:talk 3248:talk 3199:talk 3195:Paul 3172:talk 3141:talk 3107:talk 3076:talk 3049:talk 3031:talk 3006:talk 2980:talk 2958:talk 2939:talk 2922:talk 2890:talk 2875:talk 2845:talk 2812:talk 2804:talk 2778:talk 2756:talk 2708:talk 2684:talk 2669:talk 2654:talk 2637:talk 2620:talk 2596:must 2561:talk 2518:and 2491:and 2453:talk 2439:talk 2393:talk 2344:talk 2318:talk 2286:talk 2242:talk 2210:talk 2172:talk 2155:talk 2133:talk 2115:talk 2109:VMG. 2090:talk 2069:talk 2044:lift 2033:talk 2020:Math 1981:1.54 1975:4.35 1969:5.00 1958:1.20 1952:3.37 1946:4.00 1938:5.50 1935:0.86 1932:5.76 1929:2.40 1923:3.00 1915:1.91 1912:0.54 1909:2.17 1906:1.47 1900:2.00 1892:0.71 1889:0.38 1886:1.00 1883:1.00 1877:1.41 1869:0.22 1866:0.33 1863:0.59 1860:0.77 1854:1.00 1845:0.35 1842:0.50 1839:0.71 1833:0.71 1824:0.43 1821:0.54 1818:0.74 1812:0.50 1803:0.58 1800:0.67 1797:0.82 1791:0.30 1782:0.79 1779:0.87 1776:0.93 1770:0.10 1690:talk 1627:talk 1608:talk 1579:talk 1565:talk 1545:talk 1531:talk 1507:talk 1471:talk 1453:talk 1438:talk 1428:and 1409:talk 1391:talk 1366:talk 1339:talk 1316:talk 1291:talk 1252:talk 1221:talk 1194:talk 1171:talk 1148:talk 1127:talk 1108:talk 1102:you. 1093:talk 1028:talk 1009:talk 977:talk 945:talk 894:talk 871:talk 863:dead 837:talk 818:talk 799:talk 782:talk 761:talk 752:and 728:talk 703:talk 673:talk 657:talk 641:talk 626:talk 599:talk 488:talk 462:talk 440:talk 419:talk 411:WP:V 393:talk 370:talk 362:WP:V 358:WP:V 352:and 335:talk 319:talk 294:talk 255:talk 236:talk 215:talk 140:talk 136:Paul 125:talk 109:talk 2868:QED 2824:not 2747:VMG 2741:not 2729:VMG 2626:). 2584:not 2107:--> 1815:106 1794:120 1773:131 857:as 831:.-- 755:.-- 413:. 364:. 4783:) 4764:) 4744:) 4722:) 4703:) 4688:) 4670:) 4613:) 4594:) 4575:) 4555:) 4523:) 4505:) 4490:) 4466:) 4435:) 4406:) 4379:) 4333:) 4251:) 4213:) 4170:) 4134:) 4104:) 4076:) 4043:) 4000:) 3923:) 3889:) 3858:) 3811:) 3764:) 3721:) 3682:) 3653:) 3630:) 3607:) 3589:) 3576:) 3572:) 3545:) 3527:) 3508:) 3489:) 3470:) 3454:) 3424:) 3382:) 3372:do 3334:) 3296:) 3250:) 3201:) 3174:) 3143:) 3109:) 3078:) 3051:) 3033:) 3008:) 2982:) 2960:) 2941:) 2928:) 2924:) 2892:) 2877:) 2847:) 2818:) 2814:) 2780:) 2758:) 2714:) 2710:) 2686:) 2671:) 2656:) 2639:) 2622:) 2563:) 2459:) 2455:) 2441:) 2395:) 2346:) 2320:) 2292:) 2288:• 2266:, 2244:) 2216:) 2212:• 2174:) 2157:) 2135:) 2117:) 2092:) 2071:) 2058:, 2054:, 2050:, 2035:) 1949:12 1926:17 1903:29 1880:45 1857:68 1836:90 1692:) 1629:) 1610:) 1602:-- 1581:) 1567:) 1547:) 1533:) 1509:) 1473:) 1455:) 1440:) 1411:) 1393:) 1368:) 1341:) 1318:) 1297:) 1293:• 1254:) 1223:) 1196:) 1173:) 1150:) 1129:) 1110:) 1095:) 1071:. 1030:) 1011:) 979:) 947:) 896:) 873:) 851:no 839:) 820:) 801:) 784:) 763:) 730:) 705:) 675:) 659:) 643:) 628:) 601:) 490:) 464:) 442:) 421:) 395:) 372:) 337:) 321:) 296:) 257:) 238:) 217:) 142:) 127:) 111:) 4779:( 4760:( 4740:( 4718:( 4699:( 4684:( 4666:( 4609:( 4590:( 4571:( 4551:( 4519:( 4501:( 4486:( 4462:( 4431:( 4402:( 4375:( 4329:( 4247:( 4209:( 4166:( 4130:( 4100:( 4072:( 4039:( 3996:( 3919:( 3885:( 3854:( 3807:( 3760:( 3717:( 3678:( 3649:( 3626:( 3603:( 3585:( 3568:( 3541:( 3523:( 3504:( 3485:( 3466:( 3450:( 3420:( 3378:( 3330:( 3292:( 3246:( 3197:( 3170:( 3139:( 3105:( 3074:( 3029:( 3004:( 2978:( 2956:( 2937:( 2920:( 2888:( 2873:( 2843:( 2810:( 2802:( 2776:( 2754:( 2706:( 2682:( 2667:( 2652:( 2635:( 2618:( 2559:( 2451:( 2437:( 2391:( 2342:( 2316:( 2284:( 2240:( 2208:( 2170:( 2153:( 2131:( 2113:( 2088:( 2067:( 2031:( 1972:9 1688:( 1625:( 1606:( 1577:( 1563:( 1543:( 1529:( 1505:( 1469:( 1451:( 1436:( 1407:( 1389:( 1364:( 1337:( 1314:( 1289:( 1250:( 1219:( 1192:( 1169:( 1146:( 1125:( 1106:( 1091:( 1026:( 1007:( 975:( 943:( 932:" 922:" 892:( 869:( 835:( 816:( 797:( 780:( 759:( 726:( 701:( 671:( 655:( 639:( 624:( 597:( 486:( 460:( 438:( 417:( 391:( 368:( 333:( 317:( 292:( 253:( 234:( 213:( 138:( 123:( 107:( 50:.

Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.

Index