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Explain to me how a shorter rope is more difficult.


Taelan28
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Im asking for hate on this one, but I've never skied a course and I've never shortened the rope.

 

Anyways, I was wondering how exactly its harder. Ultimately with a long or short rope you still need to get from left to right at the same speed. With a shorter rope you're turning into a much bigger hole and the boat can yank you pretty hard when the boat catches up.

 

Someone please explain it to me. I know its harder but not exactly sure how.

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for the most part, its only harder in relation to fixed points - a slalom course. you have less rope to get to the required width. think of a playground swing, the longer the swing, the slower and more mellow it swings. with a shorter length, it will swing much faster. You (the skier) still needs to get to the same width as on a longer rope to run the course. Therefore, it requires more effort, technique and skill to be able to do that.
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Its a physics thing. The direction the boat is pulling the skier is always along the rope, from the skier directly to the boat. With a longer rope, when the skier is at the ball the boat is much further down course and is pulling the skier more down the course. As you ski, you are always moving down course, so this pull is in the same direction you are traveling, or at least partly in the same direction. You can easily ski much wider than the balls. With a short line length (say 39' off), when the skier is at the ball the boat is almost directly perpendicular to the skier and the course direction. So, the boat is pulling the skier directly back towards the middle of the course. This is 180 degrees from the direction you need to be going as you ski wide around the balls and 90 degrees off the down course direction. This makes it much much much more difficult to ski wide enough to get around the balls. In addition to this, with a long rope you can ski 10' wider than the balls and still hold onto the rope with 2 hands. With a very short rope, even with the skier directly perpendicular to the boat, the handle won't reach the ball. You have to release from the boat and make up that distance with your body lean. Hope this helps.
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Firstly, that question definitely does NOT invite any "hate."

 

It's a great question and it's really not that easy to answer, even for those of us who know from experience that each shortening of the rope causes a *huge* jump in difficulty of completing the slalom course.

 

I've done some physics & math modeling, and when you do that it jumps out that pretty much every aspect of a shorter rope length (while completing a fixed slalom course, of course!) is more difficult: the maximum tension on the rope goes up, the minimum skier speed goes down, the maximum skier speed goes up, the maximum acceleration goes up, the maximum deceleration force in the turn goes up, and the maximum "rip" (rate of change of force) exiting the turn goes up.

 

These increases are significant with every single shortening.

 

And yet those jumps in "statistics" are almost the easy part. It's not just a matter of handling all those increasing values, but of doing just the right thing at just the right time to actually have a chance to! The margin of error on short ropes gets very small, especially as the rope length approaches (and then passes!) the distance out to the buoy. If your ski is facing the wrong direction for just a moment, or your body is not aligned properly as a force initiates, you'll be pulled off the successful path almost instantly and then fractions of a second later the combination of your position, speed, and the current rope angle has rendered the pass impossible.

 

You'll hear a lot of mention of 38 off, 39 off , and 41 off, because many of us are trying to figure one of those out and/or dreaming about the day we might. These line lengths are all shorter than the distance to the buoy. That distance is 11.5m and the ropes are 11.25m, 10.75, and 10.25m.

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For things like this, the easiest way to see it is to explore the edges of logical extent:

What I mean by that is, what if you shortened the rope (if you could) to the point you only had 6 feet of rope behind the boat. Just imagine trying to get around the buoys from side to side at that length.

Of course this is hypothetical, but when you the shortening logic to the extreme it will immediately highlight the primary issues at play.

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Next, let me see if I can put a little more "intuition" behind one particularly difficult aspect. When the rope is very long relative to the distance out the buoy, you don't have to achieve a high rope angle at any point. This leaves you "in control" at all times -- you can choose how and when to pull against the boat.

 

When the rope is similar to, or shorter than, the distance out to the buoy, then you must ultimately achieve a very high rope angle. As you go out past about 45 degrees, you can no longer usefully pull against the boat's direction to create speed. In the context of a slalom course, this means you have a very short "angle range" directly behind the boat where you must build so much speed that your momentum carries you to the maximum rope angle you need to achieve.

 

But there's a downside to all this speed, too. Roughly 2 seconds from now, you want to be travelling in almost the opposite direction, so you gotta decelerate (in the vector sense) in a helluva hurry. And while you're doing that, you better ski away from the rope handle at just the right time with the just the right speed and direction, or else it's physically impossible to go around the buoy!

 

In the end, there's no way to fully explain it. But hopefully some of what I and others have written gives you a little bit of an idea.

 

Btw, I also sent you a "private message" a couple of days ago -- see your "Inbox" tab.

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1) As the rope shortens, the angle of the rope steepens at the widest point of the turn. The greater this angle, the greater the force that the boat's pull imparts attempting to straighten out your ski...or to move you back to center.

 

2) the speed and cadence is significantly different as you shorten the rope...it gets much faster

 

3) the required path (to get around the buoys) becomes ever more exact as the rope shortens...leaving much less room for error.

 

4) The transitions going from pull, to pre-turn to turn etc, have to be much more precise, and occur much quicker as you shorten the rope.

 

If you try shortening the rope while free skiing...you will notice the incresed speed; but, from 15 off to 28 off, it might even seem to be easier. You may experience less slack, and even a reduced effort (due to reduced drag on the ski). That said, it is a bit of an illusion, as you can turn and transition very patiently. In the course, it is a very different story.

 

After 28 off...it will become harder with each shortening...but still, nothing like when you are in the course.

 

Anyhow, try it, it can be a lot of fun.

 

 

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Ah. I see Garn. Then again the boat is moving and one should look at the trail of the ski to measure it. Purdueskier has good points. I wondered why it was so hard for me to use one hand on a turn, and now its a little more obvious, but mostly because Im lazy to learn and want to enjoy skiing at $4 a minute.

 

Bogan http://en.wikipedia.org/wiki/Advent_Rising

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Ok here's where I have to be a pedantic math nerd, even though the point Garn is making is a totally valid one in practice.

 

In about 30 seconds you'll remember why math nerds are so annoying...

 

But it's actually not quite true that the ski MUST travel a greater distance, although in practice it does. The forces in certain spots would be astronomical if trying to force a simple sinusoidal path with an extremely short rope, but theoretically it's possible. As a thought exercise, you could carve a sinusoidal path (representing the skier) and a straight path (representing the boat) into a block of wood. Then tie two sticks together at a distance that is anything more than the amplitude. It remains possible to make one stick track in the sinusoidal path while the other tracks in the straight path.

 

Ok, I've never ACTUALLY done that, and I reserve the right to have made some math error and be wrong. But I'm *almost* sure it's possible! The thing that happens though, is that the "skier" has to travel almost the entire outbound arc during a time that the "boat" barely advances at all. Then the opposite occurs -- the "boat" must advance considerably while the "skier" travels only a short distance.

 

Since the actual boat maintains a constant speed, that maps to ludicrous acceleration and deceleration for the skier -- effectively impossible in the real world.

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@Brent. Thanks. But you may want to take that compliment back after the above :).

 

Btw, several of the other explanations above are also excellent, notably PurdueSkier and Jordan. And both are a little more concise than mine. (I can do "complete" pretty well, but am not so good at "concise.")

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@Jordan - "1) As the rope shortens, the angle of the rope steepens at the widest point of the turn. The greater this angle, the greater the force that the boat's pull imparts attempting to straighten out your ski...or to move you back to center."

 

I disagree with this. In fact the force from the boat is inversely proportionate to the angle achieved. Think about this, if you have one end of a rope and your friend has the other and you stand as though he's the boat and you're directly behind the boat. He takes one step backwards and you are forced to move one step. Now go out to 90 degrees and have him take that same step; you barely move at all... He has to take several steps to move you one step.

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I think all of the above comments are good, and some much more detailed than my somewhat simple thoughts. Like many of you, I started in this addiction many years ago trying to make a 15 off pass at full speed. For much of my first 5-10 years in the sport I figued that greater energy expended (pulling harder and longer) was the way to get early and wide and get to the next rope length. Having beat the hell out of myself trying that, I finally got some coaching starting about 10 years ago and have continued that, every couple of years, since. What I've learned so far is that shorter lines are about managing speed and load, good body position, good handle control, patience and timing.

 

Now, learning each new rope length is "more difficult". However, that difficulty for me isn't in increased physical effort, but in learnig to understand when and where and how hard and with what angle to lean against the boat so that I am in control throughout the entire movement from buoy to buoy. At 28 off (my opener) I can make big mistakes, load too much, go to fast, etc, and still make the pass 99.9% of the time. At 32 off, I can still get away with quite a bit and make the pass. At 35 off, I still get away with pretty big screw ups and run the pass. At 38 off, not going to happen!

 

Interestingly, however, when I do run 38 off and run it right, it isn't "more difficult" than 35, 32, or 28. In fact, it often feels like it is easier, slower, etc. So, my point is that the shorter lines for me are more difficult, but that difficulty is measured technically rather than physically. Run correctly, I honestly use less physical effort at shorter lines than longer lines.

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@Roger wrote: In fact the force from the boat is inversely proportionate to the angle achieved.

 

True. But not what Jordan meant (or what he should have meant anyhow!). The relative size of the *component* of that force that is toward the center-line continues to increase (except that it's undefined when the total tension force is zero, but that doesn't last long because the boat will soon be pulling you again).

 

Directly behind the boat, the component of the rope's force that is pushing you toward the centerline is zero. As the angle ramps up, the "centering" component of the force becomes nearly 100% of its net value.

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I don't understand what you guys are talking about.

 

As the rope gets shorter, I get dumber. At 32 off I am a freak'n genius, at 35 off I am a journeyman and at 38 I am Frankenstein's monster running from the fire.

 Goode  KD Skis ★ MasterCraft ★ PerfSki ★ Radar ★ Reflex ★ S Lines ★ Stokes

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While David Nelson's work is *extremely* interesting, and he opened my eyes to several things, it is an over-statement to say that he has mathematically proven those things. As with any first attempt at a rigorous analysis, there is a lot more work to be done before all of his conclusions can be considered facts.
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Garn is correct.

 

Minimal geometrical distances achieved for a given line length and constant minimum amplitude of handle travel (say 65') CANNOT be lessened despite any forward movement of the pylon at any velocity (assuming a taught line along this amplitude of travel).

 

That is of course UNLESS you have 6 of these on your course, arranged in the right locations.

 

http://upload.wikimedia.org/wikipedia/commons/thumb/d/d7/LorentzianWormhole.jpg/624px-LorentzianWormhole.jpg

 

 

Do these exist in Utah or Colorado? Scott where can I get some?

 

:)

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Gloerson, you and your non-Euclidean space need to send me an email:

 

nathaniel_bogan AT alum DOT mit DOT edu

 

We may have some fun! :-)

 

Unfortunately, math and analysis like this cannot be usefully discussed on a forum. But the quickie is that Dave in that paragraph is assuming a very literal pendulum, whereas using the standard "small angle" approximation equations for a pendulum, but then forcing the parameters to match the timing and geometry of a slalom course, doesn't fail in any of the ways he describes. (It's also probably NOT the actual answer, but it doesn't fall prey to the disqualifications he describes there.)

 

I hope that some day I'll have time to publish some coherent theories. But the plain truth is that this sort of analysis is just for entertainment, so I'm not at all sure I'm ever going to really do something with it!

 

Btw, if anybody can ever get some highly sampled path data from a bunch of skiers and various line lengths, I'd be really interested. I actually would probably be a little more interested in the rope handle path than the ski path, but both would be awesome!

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Just curious- all you guys that ski at awesome short rope lengths- have you ever gone back and skied at 15 off? If so how does that feel? Is there a point at which a longer rope reduces your ability to generate speed and makes it harder to complete a pass? I was told to throw the first 15 feet of rope away- is that wrong for a new skier?
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I actually grew up with him, that is to say until he finished his 3rd calculus book in 8th grade. He was then shipped off to a private school and I didn't really see him much after that. Given the fact that he couldn't perform the basic act of a push up or a sit up(brain weighed too much) skiing would of been comical to witness to say the least. He graduated around 1993 and then started teaching there and has been a professor at MIT since then.
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@crashman - every now and then I'll do the longer line, slower speed thing, and you're correct that it is harder to run the course at slow & long, than fast & short. That's why a lot of coaches will shorten the line to -22 when teaching newbies, which gives an improvement in acceleration to help people at the slower speeds
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Taelan,

 

Good question. This is something that even very experienced skiers do not fully understand.

 

Here is a little experiment to try:

 

1. Get a small weight (marble, ball bearing, or fishing “sinker”) and tie to a piece of string 23 inches long. This represents the rope at full length

 

2. Tie the end of the string to a wall so the weight can swing back and forth like a pendulum.

 

3. Put 2 pieces of tape or mark the wall with vertical lines 11.5 inches on each side of the string. This represents the width of the buoys on each side of the course.

 

4. Gently push the weight so that it just reaches the tape on each side. Note how hard you had to push and how fast the weight is swinging back and forth.

 

5. Now shorten the string to 14 inches (28 off) an repeat. How much harder do you have to push the weight so it reaches the tape on each side?

 

6. Now shorten the string to 11.25 inches (38 off) and repeat. Again, how hard do you have to push to get the weight to reach the tape?

 

7. How much faster is the weight swinging with the shorter string than at full length?

 

 

*The actual length does not matter as long as the proportions remain the same. In this example, inches corresponds to meters.

If it was easy, they would call it Wakeboarding

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@Gloerson Who me? :-)

 

Meanwhile, I've just received a rumor of some data. Fingers crossed that maybe over this winter I'll use it to make further progress. But best guess is I won't... This sort of stuff is fascinating to ubergeeks, but doesn't lead to running more buoys. Better to spend my free time in the gym!

 

@matthewbrown Cool! I was also class of '93, so it's possible I'd recognize him. Then again, I attended an MIT reunion a little while back, and my most common reaction in meeting supposed classmates was "I have never seen that person in my life." I suppose that's to be expected with the social "skills" of MITers.

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@Taelan28, what I can tell you as well is the "physical" vocalizations and thoughts of experienced open water skiers the first time in the course.

 

"I usually ski @ 36MPH in open water, let's go for that"

"Well, yeah, if you insist I can try going @30MPH the first pass"

"Why shouldn't I try to go thru the gate?"

"The guys in the boat do not have a clue, I will show them how it is done"

"Man, this is sloooow..."

"Here comes the gate, oooops..., well, does not matter, next time"

"OK, let's kill that first buoy"

"Hey, that damned buoy is waaaaay too wide!!!!!"

"OK, BIG TURN!!!!"

"OUCH, my back..."

"Ok, now to the second buoy!!!!"

"Where the hell is the damned buoy!!!!"

"Here it is!!!! Wait, looks like this is the fourth one..."

 

 

 

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36 long line... a pretty darn hard pass.

 

It been a few years but a couple times a summer, after a day at the lake, we would have 36LL challenge. No warm up, off the dock, 2 passes down the lake & back, spinning at 36LL to see how many buoys you could get. You'd be amazed to watch some very good shortline skiers flail at this pass. If you try it make sure to fire up the video camera as it can get quite comical.

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So from my standpoint, we could probably use all of the math terminology you guys just mentioned (which by the way, made me feel like I didn't pay enough attention in Trig and Calculus) to explain all of the nitty, gritty details. I wish I understood all of what I read, and I know there is a lot of great info in this thread. I would like to add my humble, and hopefully simple opinion.

 

One of the main differences is that when the line gets shorter you must travel up to a higher point on the boat to achieve the same width. In other words (as someone might have touched on earlier in the thread) you must travel further around the arc created by swinging back and forth behind the boat. The center of the arc is, of course, the pylon. This does NOT mean we travel more distance (the radius of the arc is actually shorter therefore so is the circumference of what would be the circle that represents the arc), but it does mean we move up to a point where the boat isn't actually pulling us (when we are almost directly beside the boat which happens as the rope gets shorter). This means that our speed, direction, and timing become increasingly more critical as we shorten the line. IF you arrive at the buoy with too much down course speed, you must in essence wait for the boat to "get out of your way" before you head back to the other side. This scenario creates a big slack hit as you decelerate and wait for the boat to pull you. This is where the physics of the swing meets up with the timing, and create harsh demands on the athletic skier.

 

In other words, rhythm and timing are very critical and when they aren't correct we must be "more perfect" mechanically on the ski in addition to being more fit.

 

This is the quick and simple version of my thoughts on the subject. I try to keep the thought this simple so that I can keep my own goals centered around the things that have the most impact. For me it's timing/rhythm. I can be as mechanically sound as possible (for me), but if I pull too long I generate tons of extra down-course speed and therefore over shoot what should be the natural apex of the turn. This creates slack which causes my skinny frame to crumble and lose position and therefore power and speed coming out of the turn. At this point every turn has slightly diminishing returns.

 

Just my thoughts...

 

Seth Stisher

H2OProShop.com and SethStisher.com

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@Sethski Excellent feedback as well as what @Bruce Butterfield posted. Try his model but use a pencil to draw the path of travel for the assumed minimal amplitude (e.g, 65’ to scale). Now shorten the “rope” (lessen the radius), yes the circumference decreases but one is now traveling a greater portion of that circle when the line shortens (as you adroitly stated). Measure the actual lengths of the paths scribed as the “rope” shortens; the distance along the arc traveled INCREASES as the “rope” shortens. Assuming a similar intercept point at width on each side relative to buoy position; there is NO way this minimal geometrically (Euclidean @ThanBogan) determined length can be lessened.

 

However, the actual path of the handle can be increased dependent on technique; those who ski very wide & early at longer line lengths will move that intercept point (with the minimal amplitude line) further in front of the buoy (up course) and the handle will travel further than those skiing “coordinates” (hesitated to use that term). In these instances a longer path will be skied than minimally required and as they shorten the line the increases in distance & speed may not be as perceptible (assuming they mastered the technique of moving their intercept point further down course without detriment when shortening the line). Of course it’s a basic tenet that the “path” at ultra short-line becomes highly defined; who was it that stated it’s like skiing a path as wide as a garden hose? Nonetheless, assuming the same “intercept point” at full, minimal amplitude (referring to handle path); as the line shortens the minimal distance of required handle travel INCREASES.

 

Below some basic trig review:

Arc length = Ropelength x arc angle in Radians.

 

Assume the following (along the lines Seth described getting further up on the boat):

35 off arc angle = 120 deg

38 off arc angle = 138 deg

39.5 off arc angle = 152 deg

41 off arc angle = 180 deg

 

There are 2π (approximately 6.2832) radians in a complete circle,

So:

2π radians = 360 deg, 1 radian = 360/2π = 180/π.

 

Therefore:

180 deg = π Radians or 3.1416 Radians

35 off arc length = Ropelength(40) x 120/180 x π = 84 feet

38 off arc length = Ropelength(37) x 138/180 x π = 89 feet

39.5 off arc length = Ropelength(35.5) x 152/180 x π = 94 feet

41 off arc length = Ropelength(34) x 180/180 x π = 107 feet

 

It’s all interesting stuff, although not helpful to me in skiing shorter lines other than providing a tangible explanation of why I find it so much more difficult.

 

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Gloersen hit on one of the key factors, the fact that the handle travels a greater distance as the rope gets shorter, notice the difference at pretty short lengths, as the curve flattens for the longer lengths that distance will get shorter. Couple that with what Seth commented on, the fact that at shorter lengths one has less "useable" pull (a perpendicular rope to the boat offers the skier nothing to assist the pass, parallel to boat offers excellent pull) creates a greater challenge to optimize the pull in a much narrower window (directly behind the boat) and Than's comment on the effect of needing to reverse direction exactly when the rope is actually accelerating the greatest amount in the linear or "forward" direction or more and more when the rope is shortened simply are several elements coming togehter to make the shorter rope length pass more difficult. Notice I did not really include the skier path here as there is some level of independence of it and the handle path (in fact slack adds another factor of independence or complication to modeling the skier path). Fun stuff to ponder on a winter day!
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Seth,

 

As Gloerson calculated out, the fundamental difference is that as the line gets shorter, the skier's path IS longer. While most of the other points are true, this is the fundamental reason why shorter lines are harder – you are traveling a longer distance in the same amount of time, therefore your average speed HAS to be faster. Generating more speed requires more tension on the line. There is no way around this fundamental point of physics.

 

Of course generating this extra tension and speed efficiently (feeling easy) and staying in control are the hard part. The rhythm, timing and mechanical alignment are critical to stay in control, but if you can't / don't generate more speed to begin with, none of that matters.

 

For those who's heads start to spin (or fall asleep) with the math, spend 5 minutes and try my little experiment. If a picture is worth a thousand words, that exercise is worth 10^6 words from the math geeks.

If it was easy, they would call it Wakeboarding

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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.

 

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I'm with Seth. At shortline you have to get higher on the boat (right next to it at extreme short line) to achieve enough width, and max width then also has to come later just to get around the ball. The turn-in back toward center also has to be later.

If the skier is next to the boat and turns in immediately, the first travel is directly at the boat causing a slack situation that is unrecoverable. This is the "patience" required to, as Seth mentions, let the boat get out of the way. Ever notice how far down course the great short line skiers actually break the buoy plane and head inbound? Very different than finding an early apex at longer line and coming back toward the wake off the backside of the ball. This is only possible at these lines b/c the skier is still behind the boat, not next to it when rounding the buoy.

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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

 

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I do not really follow all of the math here but this thread is an example of what i hoped for when I stared my first forum. I see that every one does not agree and that is the way it should be. The point is smart guys are exchanging ideas about the sport we all love. Sharing ideas is the catalyst to greater understanding. All ideas are welcome.

 

what the average reader may not know is the brain trust that is on this site. Besides the occasional elite skier like Seth we have MIT PhDs, NASCAR engineers, civil and mechanical engineers, pilots and a old used car broker. Plus the staight sports geeks with masters degrees in things like exercise physiology and physical therapy. I am sure I have missed a few key degrees/ smart guys but you get the idea.

 

when I say pilot, I mean smart fixed wing guys. No one cares what helicopter guys think

 

I just wish I knew an elite skier with a degree in civil engineering.

 Goode  KD Skis ★ MasterCraft ★ PerfSki ★ Radar ★ Reflex ★ S Lines ★ Stokes

Drop a dime in the can

 

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