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Horton - could you elaborate on the conversation you had with Mapple? Sounds interesting. Marcus - the section of your post Horton quoted is undeniably true, but could you explain what you'd suggest skiers do to use force efficiently and how you put your body where/when it needs to be? I'd like to quote another part of your post, that "the more theory and mechanics (correct theory and mechanics) you understand, the higher you're potential on the water....and the more you will ENJOY the ride." All - I agree with comments by many of you that math has little meaning to you. That's why I never post any math in anything I write - I only mention the general approach I take with the math & physics. But showing the results of that analysis, like the skier path, speed, and force at each point in the course is the kind of thing I can directly relate to my skiing, and I hope you guys can too. If I'm just wasting your time with it, let me know and I'll keep my thoughts to myself.

One more thing – something I've got to respond to directly: the idea that a shortline ski path “MUST be longer” because the arc length drawn with a short line rope is longer than the arc length drawn with a long line. I agree, the arc length is longer at short line. I agree, “there is NO way this minimal geometrically determined length can be lessened”. But this does not prove the ski path is shorter or longer because it ignores other important information . The math supporting this contention (consisting of a couple numbers multiplied) is not the problem – the multiplication is correct. This math proves nothing. The issue is the wrong problem is solved! What’s wrong? People claiming arc length sets limits on ski path are guilty of mixing their frames of reference. (bait and switch is another way to think of it – not that I think this was done intentionally – it’s just that frame of reference issues are always tricky) They use the pylon frame of reference to measure distance and the lake frame of reference to measure speed. If they want to be consistent and use the pylon frame of reference for both distance and speed (while ignoring the speed of the pylon moving down the lake) I’ll agree the arc length is longer at short line, but are they willing to agree with the skier speed you’d measure from that frame of reference? The speed of the skier advancing toward the boat will vary between 0 and 12mph. The speed in the direction perpendicular to the boat path will vary from 0 to 37mph, for a typical 34mph skier. So what? How useful is it to realize that at some points in time, with respect to the pylon, the skier is standing still? What we really care about is how fast the skier is moving across the lake, because that’s what determines where we are and how large the forces are. And no, you can’t take the arc length and add the boat speed to it independently. Data from two different frames of reference doesn’t add with simple arithmetic. Let’s be consistent: if the speed with respect to the pylon is meaningless then so is the arc distance traveled with respect to the pylon. The only frame of reference that’s meaningful to the way we ski is the one with respect to the lake. From that frame what’s relevant is the distance between buoys and, if you want to use a pendulum analogy, the rate of change of angle the rope makes with respect to the centerline as the skier loads the line. The ski path distance varies tremendously as the angular rate of change is varied, regardless of line length. The minimum path, even at 39, can be the same path a skier takes at long line. Schnitz and many others have proved that skiing his coordinates style between buoys. Still want to claim the shortline path MUST be longer? If you’re going to use a pendulum analogy that is relevant for ski path distance, speed, and acceleration you have to simultaneously look at boat speed & rope length, which are independent of the skier, and the angular rate of change the skier chooses to create by how he loads the line and where he places his center of mass. If you want to see some ski paths that are mathematically correct & based on physics, look at my article ‘possible and impossible lines’ on Schnitz’s web site. If you want to see some ski paths measured from first rate video with 30milli-second time resolution look at my 2006 article on the same web site. Nobody’s disproved those articles yet, and a lot of people have written me trying. In most cases, they’ve ended up agreeing with me. I know – some people still won’t understand, still won’t be convinced. That’s just a fact of life. You can’t prove everything to everybody. I suffer from the same problem: Albert Einstein himself couldn’t PROVE his theory of general relativity to me: not because it’s not right, not because he doesn’t explain it convincingly, not because data is lacking, but because I don’t understand Reimann geometry or tensor calculus. I regard that as my problem, not his. So I can choose to trust someone smarter & more knowledgeable than me, or not. I choose to trust him. Similarly, I could produce an exact mathematical proof of this ski arc thing, but what’s the point? Those who understand what I say are already convinced – probably were so before I ever said anything on the matter. Those who don’t understand the points I’m making certainly won’t understand the math involved. I’m writing this to the guys in the middle who don’t know whether to believe what they read or common sense. There are a lot of good comments on this thread - the ones by Razorskier and Seth in particular. David Nelson

I've just had email from several of you asking me to read this thread and respond to it - first I'll address my critics: Than makes a valid comment that Scott's quote of me is not 100% accurate because it uses a small angle approximation for a pendulum. That quotation was in response to someone making an inaccurate claim based on the small angle approximation, so I just used the same approximation to point out the error of their claim. If I had invoked the math that avoids the approximation few would have understood and my answer would have been blown off. So although what I said was not quantitatively true, it was qualitatively true. Now, I'll warn Horton & a few others to punt now and skip to the next paragraph if you don't want your brain to start sizzling. A year or so ago I did do exactly what Than suggested was possible: I did use a Taylor series expansion of Legendre's first elliptic function to solve the full angle pendulum equation and force the period to be equal to the time a skier has between buoys. Then I modified the rate of change of angle, which is under a skier's control, which is how I came up with the plots in "possible and impossible lines" on Schnitz's web site. You may want to read that if you haven't. It is pretty concise in stating what makes shortline harder. I don't spout any math/physics in that article, but I assure you it is there and it is correct. So pendulum motion can definitely be applied to skiing, but the simple analogies most people make to pendulums lead them to incorrect answers, including that your path must be longer on shortline, that speed must be higher on shortline, that force must be higher on shortline, that acceleration must be higher on shortline... However, I'd agree that for most skiers, those things do happen at shortline and that is why it gets so much harder. Those things don't have to happen, but they will happen if you approach shortline the same way you ski longer lines. I noticed, and measured accurately, when most pro's ski shortline they apply forces differently as the lines shorten, and those things don't happen. Doing something differently to make sure those things don't happen is the key to skiing better! In private, several pro's & Big Dawg skiers I've talked to agree they have to approach shortline differently, but they regard that information as a 'trade secret' and they're not about to share what they actually do with me or on a web site. They do say my physics makes sense to them. Others I've talked to agree something is different at shortline, but they've never thought about it much - they just respond differently at shortline to make the pass. And yes, I do know some great skiers who approach shortline the same way they ski longline, and they do run the pass and their speed-distance-acceleration do increase. It's not impossible to ski that way, just harder. You can try smarter or harder, but as in most sports, smarter gets you further. So why don't I run 39 if I know so much? It's been shown, in sports or any other discipline, it takes 10,000 hours of practice to become an expert. An article in Waterski years ago pointed out it takes on average ten years of skiing ten or more sets a week for someone to reach pro status, assuming they are athletically 'gifted'. I'm not 'gifted'. I started skiing at age 48 and in a good summer I ski 100 sets and I'm 59 now. So I've practiced less than 400 hours. I think correctly understanding physics can shorten the learning curve, but since translating brain memory into muscle memory still takes a long time, I'm not going to make it. Also, by the time I figured out I needed to do something differently at shortline my reaction times were slowing down, so I can't make my oncourse angular rate of change match what the physics tells me to do. But I'm happy with my progress. And I have coached a few people who are younger and who have made a lot more progress with my approach than with what they'd been doing before. Wow - I'd really rather be skiing than writing this! And although I am an expert at snow skiing and snowboarding, because I do have my 10000 hours there, I can't get to the slopes everyday like I did when I lived in Utah. At least in the summer I can ski most days in Colorado.

Bud asks does speed create angle. If you advocate that, you'd better ask what created the speed to begin with. I have not observed any skier where that is the case. I agree with Bud that angle creates speed. The skier comes around the buoy with with some speed and angle, consciously increases the angle of the ski, and that causes speed to increase. Then in the preturn speed can be used to maintain outward direction to the extent rope and boat position will allow. I don't think anybody disputes that simple logic, yet I think I understand what people mean when they say speed creates angle, but strictly speaking, that's wrong. Angle is the cause which translates the force supplied by the boat into new direction, and speed is the effect, not vice versa. Now on to the question others raised about whether you can add buoys by skiing with more angle and higher speed. Definitely! That's what my wide-early chart is all about. If your present limit is 38 off, and if you could muster more angle and speed, you could move up to the 39 off pass, etc. But that begs the question: if you're skiing that style and your limit is 38off, it's because you've reached your limit - so what makes you think you're going to be stronger or have faster reactions? For most of us it's wishfull thinking. Of course we could all do better if we were stronger and faster - but we all do have limits. So the question should be: how much better could you ski if you made better use of the strength you have? At first, you're likely to ski worse when you try a new style, because so many of your habits and reactions will be counterproductive. Tiger Woods didn't immediately improve when he took a few months off to change his swing many years ago, but he had faith, didn't get discouraged that he initially got worse, worked hard, and came back with a vastly improved swing. If the approach of skiing smarter not harder makes sense to you, then maybe you'll persist until you learn the new style and practice it enough to where you can match and then exceed your old style PB. Last year when the Big Dawg came to my ski site, I found myself a perfect spot on the shore to sit and watch all those great skiers. Most of those guys don't say or perhaps know what style they ski. Most of those that will talk about it advocate a wide-early high angle high speed approach. So I watched them ski. At their 32off opening pass, judging by where they crossed course center, the majority of them looked like wide-early skiers, crossing well upcourse. Those that were crossing near midcourse even at 32off were clearly not having to work as hard, however. Then as the line shortened, the wide-early guys were crossing later and later, while the efficient skiers just kept crossing near midcourse and making it look effortless. At 39off they were all crossing within a meter of center. So even those that think they ski a wide early style were not. Even those guys don't have the strength and reaction time to take that style into very short line, so what chance do you and I have with a wide early style? I see the same thing watching the pros, except very few of them even start with a wide early style at longer lines. These observations, along with the forces I compute for the wide-early paths in my charts, are my justification for saying nobody is strong enough to ski all passes in a true wide early style. They may still be giving it all they've got and attempting that style, but why do that if somebody else can do as well with a lot less effort? If I learn to ski with less effort, I expect less wear and tear on my body, and thereby assure myself a lot more enjoyable years behind the boat.

thanks Bud - the graphs are not drawn, they are plotted directly from the math, as you thought. You asked what a skier needs to do to keep the speed balanced. If you do as you suggest and load the line less but longer, you will cross the centerline later. That will bring you closer to the efficient path, and your speed will be more balanced. I don't think you can keep crossing up course and accomplish it - we can choose our actions but not the consequence - at least that's what I tell my kids. ... skidawg - this is something I do to relax, like some people watch tv. It's pretty tame compared to what I do for work - of course, when the water gets to 50F in a week or two, that's where I'll be doing my relaxing for the next 6 months.

Bud got me thinking again about skiing strategy a month or so ago. I've been an advocate of efficiency, but that's still pretty hard to define. I've discouraged the wide-early style, but I've never been able to say clearly why or explain it in a way that was very convincing to its advocates. The attached pdf file is my latest attempt at explaining both, and I put forth a compromise ski strategy that may make more sense to both sides of the issue.

I'll work on putting together something this evening - what I've sent to Bud piecemeal and with the help of phone conversations needs some work to make sense on the thread.

can anybody tell me how to make the jpg file appear with the text?

Following the discussion "ZO Analysis", I wanted to address more closely the issue of how ZO may affect different skiing styles. The two styles I'd like to consider are what I'll nickname efficient and wide-early. The efficient style is the one I talked about in my articles on Schnitz's web site or the possible line Chris Rossi talks about in "Skiing the Impossible Line". The wide-early style is the one that so many of use default to, and even prefer to use, because it works so well at longer lines - but at short lines it evolves into the "impossible line".  One popular version of the wide-early style is to complete your turn, immediately load hard against the boat, and maintain that max load all the way to the 1st wake, then prepare to edge change. The efficient style recommends lightly loading against the boat and progressively increasing that load to the 1st wake, then prepare to edge change. The wide-early approach will lead to crossing the boat path several meters up-course of the progressive approach. It will also give you higher speed at that point.  Both those can be translated into being a lot wider and earlier than with the progressive, efficient approach. The best the efficient approach can do is get you to the next buoy 'just in time'. As I said in 'ZO Analysis', I think the way ZO responds to load makes the wide-early approach a lot more difficult than PP or manual driving does.  Like I said there, my understanding of ZO is based on my observations as a skier and driver, and not on detailed measurements of the way ZO responds or on insider knowledge of the way ZO is designed - so I may be off base in some or many respects. But using my understanding of ZO I created a computer simulation showing how ZO responds to the wide-early loading and the progressive loading of an efficient style.  Refer to the plot I attached (I'm not sure where it will appear relative to the text.) The time scale goes from when you pass one buoy (0 seconds) to when you reach the next buoy (2.67s later at 55kph). You cross the boat path somewhere in the vicinity of 1.2-1.3 seconds. The top graph plots the skier load versus time. 800 pounds of load is what I've heard some say has been measured. So in the blue plot the load ramps up to 800 and stays there for 300ms. You'd have to be pretty strong to do this, but I know some of you are. In the progressive case, the red plot ramps up more slowly and doesn't peak till the 1st wake is crossed, with the max load there being about 500 pounds. Then in both cases, as you begin the transition to the preturn, the load drops off by the time you cross the 2nd wake. (This is based on a 180 pound skier, so if you assume the same skier strength to weight ratio but a lighter or heavier skier, you could just scale all these plots by the ratio of the skier you assume.) The bottom graph shows the how the boat speed changes in time to each load with ZO in control. This is based on my estimates of throttle response time, engine rpm slew rate, boat interia, etc., and a typical skier setting (B2?). I'm sure ZO engineers would tell me I've goofed somewhere, but I think as a rough model this is useful. The blue line is the speed response to the wide-early loading. Boat slows down 0.6mph, then speeds up and peaks at 0.5mph above the 34.2mph average. The area below 34.2 equals the area above 34.2, so the boat average doesn't change. The red line shows the progressive-efficient boat speed. The slowdown and speedup are only 0.3mph, or half as much as in the wide case. I know when I ski wide-early, the boat really seems to fight me and my buoy count drops.  When I load progressively and take the efficient path, the boat response seems benign, and my bouy count climbs. The difference in boat speeds based on this computer simulation is not as dramaticly different as I was expecting, but maybe it is different enough to explain why it feels so different and effects me so much. It doesn't take a lot of intereference to mess up a pass.

scoke - very interesting comments: I didn't realize PP had versions that controlled the speed so closely. Doesn't surprise me it made skiing difficult, especially to people who were used to manual driving. Like I commented to Horton, if we had a really tight speed control, it wouldn't stop our complaining.  I've noticed when I drive manual or ski manual, the speed vs. time plot is more like that of present day PP, and my friends & I tend to do better with it (we've got the option of ZO, PP, or manual at our club). smanski - you guys that love ZO should educate the rest of us what is wrong with our style! How do you describe your style? Do you stay 'invisible' to the boat, or do go completely the other direction and make use of the hard 'payback'? skibug & jody - makes sense to put an accelerometer in the loop to remove the GPS delay, especially since the GPS delay relative to the pull could be as short as 0 to as long as 50ms, depending on when the last reading was. Then just use the GPS for buoy to buoy accuracy. acmx - I can't say I have any new or refined slalom theories since I wrote that in 2006. Wow, time flies. I used to be more active in the BB's, then got too busy with other things for a few years. I'll see how it goes now.

Yeah, I'd love to hear what Will Bush said too! Write him a letter & see if he'll write it down. You asked if a faster response from the boat would help things - YES. When the boat takes 200ms to respond, you'll slow the boat say 0.5mph before ZO & the engine starts to reverse the loss. If it could respond in 2ms, you'd only have slowed the boat 0.005mph before it starts to respond. You wouldn't notice that little of a slowdown. As far as you'd notice, the boat would respond equally and proportionally to your pull in real time. There would be no 'payback' later! The boat would always go 34.2mph and we'd have to find something else to complain about. The problem, as I mentioned earlier, is that for a variety of reasons we're unlikely to ever get the response much faster than what it already is. Besides the built in limitations of physics, the ECM has to worry about emissions, fuel efficiency, engine stress & wear, etc. and then there are limitations on how fast GPS can update too. The next best approach is to anticipate the skier pull, with the overall goal to be the same as in the first case: keep the slowdown minimal so the payback can be minimized. If ZO can be improved at some point, great. Until then, work on becoming 'invisible'. good comments, Todd. David Nelson

Good question skiergeeg I observe the rpm surge as soon as the skier starts to load, usually just after they get both hands back on the handle. The rpm surge gets the boat back to 34.2mph prior to the first wake, and the higher speed payback starts immediately or soon after. On something like an A1 setting the correction continues into the preturn quite a ways. On C3 it ends sooner. It never lasts into the turn. On C and especially C3, the rapid/strong correction causes me to be way early & wide - then I usually waste it by standing too tall in the turn. I find I'm better to be just in time than early, so I do better with B2, but I keep trying other settings because I'm never satisfied. To ski ZO well, you need to try and stay invisible, meaning: load progressively and get an early edge change.  By loading only lightly early on you won't be slowing the boat a lot. Then ZO will only have to engage lightly. As your load increases toward the wake, ZO is already engaged and the correction will also be progressive, hence nearly proportional! That's not disruptive. If I could do that I'd be skiing like Andy Mapple, for whom ZO was optimized in the first place!  Only in my dreams. It sounds like you're skiing closer to that ideal than most of us!Â

Thanks for the comments – I’ll make a few specific replies: I agree that the ZO “making up for it†phase is what makes it harder to ski behind. ZO probably has a lower standard deviation from 34.2 than PP, but to achieve that it accelerates and decelerates in a disruptive fashion. PP has bigger excursions around 34.2 and they last longer, but that's easier for a skier to deal with, because a skier response time is about 240ms! We don't react well to things that change faster than that. I don’t know what ZO update rate is specifically, but most GPS units update 20 times/second, hence I’m assuming 50ms. The 120ms throttle response was for a BMW S62 5L engine used in M5’s and Z8’s. That is actually the response time from the accelerator pedal to ECM to DC servo motor that controls the fuel injection. As soon as that fuel enters the cylinder the engine will respond in under 4ms at 3600rpm, and then it takes some time to rev up the rpm.   If you’ve driven a car like that, you’ll swear there is no lag, but that’s because the human response is considerably slower. I too have been too lazy to video the gauges and try and correlate what the boat does with what the skier experiences. I bet ZO already has this reporting function built-in to communicate via USB, but I’d be surprised if ZO would ever make it available to us. I do agree ZO is just reacting. I don’t see any indication ZO applies the knowledge of where it is in the course or what the skier history is to anticipate any kind of pull – too bad it doesn’t.  For example, if the skier misses a pull, the engine is not throttled in any kind of anticipatory fashion. Adding this predictive type of response is something I think ZO should do to reduce the slowdown and speedup that is so disruptive to most skiers. I don't think it would be hard to add - I hope ZO agrees. One other option would be to keep the reactive approach but make the throttle response MUCH faster so that in reaction mode, the skier can't reduce the boat speed so much as they can now. Then less 'payback' would be required after the pull. I'll bet ECM could be designed to respond faster, but since existing ECM is designed to work well with a human in the automotive feedback loop, ECM should be re-designed for the way ZO is trying to use it. Given all the emissions features etc. ECM is required to control these days, that may be too much to ask, however!  If that can't be done, the anticipatory approach will likely work better. Thanks for the fact sheet HO410. I think my description of ABC fits reasonably well with ZO’s description. My 123 description seems to vary from theirs, but my experience seems at odds with what ZO claims. Since in all cases, be it a heavy or a light skier, skier load slows the boat enough that ZO has to respond ASAP to prevent the error from getting too large. To say it takes a light load or heavy load to initiate an rpm response makes no sense in my observations. I chalk up the ZO statement on 123 to tech-to-journalistic translation error – something I see a lot of in almost all technical publications. David Nelson