What changes when the rope gets shorter? Why does it get harder? How does that relate to speed?

[mrpreuss]

As the rope gets shorter you must travel further to reach the buoys.

 

If you have the same amount of time, you must have to travel faster ......?

[Gloersen]

I know this will incite a plethora of debate, but fact remains, regardless of the pylon moving forward, if the handle travels a minimal necessary amplitude for all line lengths, as the rope is shortened the path lengthens within the defined amplitude. The math is simply the formula for determining the length of an arc. Plucked from a comment elsewhere:

 

For a given (constant) amplitude, the distance traveled by the handle from "buoy line to buoy line" increases as the rope is shortened (given constant amplitude). Certainly there is a minimum amplitude the handle must travel regardless of and for each line length, thus as the rope is shortened the handle must travel a greater distance through this "minimum" amplitude. Greater distance traveled in the same time equals greater speed.

 

What happens outside this "minimum" amplitude or buoy line is another story applicable more to the distance traveled by the ski rather than that of the handle.

 

Regardless though, the skier must travel the same speed as the handle along the path of "minimum" amplitude, a path that lengthens with shorter lines and thus accounts for the greater speeds detected as you pointed out with shorter lines using the radar gun on AM.

 

[Razorskier1]

I don't find I have to travel faster within my range of ropes, which currently run through 38 off. Also, I like to think of the passes as different, not necessarily more difficult (when skied correctly).

[501Brandon]

@Razorskier... I like that thought and think I will try to instill that mental note. Now I just need to find the differences between the passes and learn the niches of those passes...

[6balls]

None are difficult when skied correctly, eh? A 38 pass can feel easy, but hard to duplicate. You need to ski more consistently correct as the rope gets shorter, and need to know what "correct" means.

I have much more of the necessary physics in my head now than ever before which drives technical changes in the hunt for shorter lines...sure wish I'd figured it out younger. My brother and fellow ballers have been a great help. Now the challenge is taking the principles in my head to bodily execution.

[Than_Bogan]

This question turns out to be very hard to answer. Almost every simple analysis is woefully inadequate. My own analysis is also inadequate, but I think I've gotten much further than most. If there are any (other) hardcore math geeks out there, let me know and perhap you and I can eventually put something together worth publishing.

 

However, it's not clear that there is any USEFUL insight from understanding exactly why shorter ropes are harder. The more useful stuff relates to the techniques of how to overcome the challenge.

 

Still, Math Guys like me can't resist trying to understand things just to understand them. Without going TOO far into the arcane: The path length question is one of many that is not trivial. It is NOT physical mandatory to take a longer path on a shorter line length. That claim is based on an assumption of a particular form of the function that describes the rope's angle [usually the "simple pendulum approximation" of theta = sin (omega * t) where theta is the rope's angle, t is time, and omega is a constant that makes this path fit the geometry of the course].

 

That said, I *think* it turns out that the simple pendulum assumption is actually fairly close to the optimal, especially in the neighborhood of -35 and -38. Trying to force the path length to not change results in ludicrous accelerations at certain points (and I do mean ludicrous).

 

As far as I can tell (but it's hard to prove because of the infinite number of possible paths for any given rope length), every rope length requires greater acceleration and deceleration, greater maximum speed and lower minimum speed, greater peak rope tension, greater peak turning pressure, greater peak "rip" (first derivative of force), and is just generally more constraining in every way.

 

But like I said, knowing that doesn't seem to be any help toward running more buoys.

[Than_Bogan]

If anyone is interested in helping push my analysis along without doing tons of hairy math, how would you answer this question:

 

** What characteristics would you be looking for for the optimal path? **

 

Do you want the peak force to be minimized? This would basically limit the strength that was required to do it.

 

Do you want the peak "rip" to be minimized? This would limit the "abuse" on your body.

 

Do you want the *average* force to be minimized? This would basically limit the total effort required to do it.

 

Other?

 

And finally, if you're sitting there reading this and thinking "Doesn't the Euler-Lagrange differential equation allow one to solve for optimal paths?" then you better contact me right away! nathaniel_bogan AT alum DOT mit DOT edu

[skidawg]

add, traveling faster means you have longer to spend in the turns (out and in)

[Razorskier1]

Than -- for me optimal path seems to be as slow and soft as I can lean while still achieving 'correct' width. Up through 35 it feels like not much of this matters -- there are about 1000 different ways to run through 35 off -- fast, slow, too wide, too early, slam the turn, soft turn, etc. At 38 it feels like being on the right path matters more, although I am sure for the guys who are really consistent at 38 they can get away with a lot of variations there just like I can at 35. As you noted, any short answer is totally inadequate, because every piece of the puzzle has to fit together. For example, if I say to lean lighter, but you give up the handle too soon, it won't work. I have been trying to boil my 38-mindset into as few things to think about as possible, and have come up with the following.

1) ski tall

2) light on the handle in the turn in

3) stay away from the one ball

 

For me this translates effectively. By skiing tall I maximize my leverage and don't give anything away to the boat by being overly compressed. By being light on the handle, I stay more upright in the lean all the way from turn in to edge change, reducing my maximum speed, but maintaining a more controlled speed (probably similar average to a 'speed up, slow down' style). Staying away from the one ball keeps me on the handle and with light outside edge pressure longer as the ski is slowing for the turn, and typically what I do at one carries through to 2-6 (which is why I concentrate on staying away from the one ball). Staying away from the ball also keeps allows me to set up to turn after the boat has had a chance to advance, rather than rolling edge too soon and advancing on the boat too quickly, then being left with no line support coming out of the turn.

 

As you note, it isn't this simple, because lots of other pieces have to fit together too. To your questions above. I want to minimize peak force, minimize the "rip" (let the ski run a path once I've stayed away from the ball), and minimize average force. This combination is easier on my body, gets me down course or out the end gates at 38, and lets me practice it more.

[Roger]

@Razorskier1 - ever had the speed gun on you? I know that at least a dozen skiers have been measured with the speed gun and a few with a system Dave Benzel used a number of years ago. All the skiers measured skied approx the same speed from 22 through 35 off; at 38 the speed jumped up considerably for every single skier measured. I believe Wade Cox commented after one of these sessions that he thought that's why so many skiers get stuck at 2 @ 38. I wish I still had my radar gun...

[Razorskier1]

@Roger - nope. I may be faster just because of the geometry, but what I try to do in my lean is to really stay off of the throttle. When I turn in and start to accelerate I lighten up almost immediately rather than sticking with it and gaining more speed. So . . . It is possible that the speed is higher by default, but what I try to do is not generate any more speed than I need. Radar gun would be interesting.

[Than_Bogan]

Be wary of the speed gun data. There are at least two problems with this sort of data:

 

1) If my current theories are pretty close to the truth (which they may not be), then there is very little variation in speed behind the boat on the optimal path, regardless of rope length. The much higher speed is achieved somewhere just beyond the second wake, as the geometry of the rope swing is now adding to the forward component of your velocity. The minimum speed occurs in a similar place at the end other end (exiting the turn), when the rope swing is subtracting from the forward component of your velocity.

 

Bottom line: You need continuous speed data, not just sampled speed data.

 

2) An actual skier doesn't use the "optimal" path on their easier passes, because by optimal I mean least effort. What an actual skier wants to minimize is the chance of failure. This typically means using more effort than necessary. This equates to concepts like getting wide and early and creating a margin of error with a more aggressive line. These are great strategies, but they confuse the issue of measuring an actual skier's speed. Because I'm doing more than I actually have to to run -28 and -32, my measured speeds are likely to be closer to -35 than they have to be.

[Than_Bogan]

@Razorskier1 The types of points you are making are the ones that are actually valuable to improve! If anything, you want to "resist the geometry" and not give in to its inherent tendency to give you more speed.

 

That is why I consider the "Why is this harder?" question to be of purely academic interest. Of course, I am an academic, and I am interested!!

[Than_Bogan]

@Razorskier1 From a math perspective, you can't choose to simultaneously minimize 3 variables (peak force, peak rip, and average force). You either have to pick one or come up with some way to weight each of them. Ultimately you can only choose to minimize one thing. Of course, it's *possible* that such a minimum will, by coincidence, also be the minimum of the other things, but you can't force that to be true.

 

(If it were to prove true, however, it would be a shocking elegant result that would make it almost inarguable what the optimal path looks like. Usually math is not that kind, though!)

[Razorskier1]

Than, the only variable I really think about is what the lean through the gate feels like. I try to minimize the energy I put in behind the boat. In the "old days", I would hit the bottom of the pendulum and wail on the boat, generating both enormous load and too much speed. Now, when I feel the line start to move up the other side of the pendulum, I just try to act like a weight on the end of the line, not adding pressure, nor taking pressure away. This 'feels' to me like mroe of a constant speed strategy than a speed up/slow down strategy. Another way of putting it is that I feel like I am 'letting the boat win' when it starts to want to pull me up, rather than staying on the lean.

 

Your other comment about optimal path at longer lines is something I have been working on -- take my -38 mindset and translate it to longer lines. Easy to do at home, sometimes harder to do at tournaments. Why? Because of what you said -- I tend to put a little insurance policy on the tournament -35 to ensure that I get my shot at -38. Meaning that I ski it with more energy than I would in practice. In the last week or so that bell went off in my head really loudly, and I have been very disciplined in practice about skiing the optimal path regardless of line length. It has been much lower effort, and it has improved my consistency in getting down the course at -38 beyond anything I've done before. This has required more focus than one would think, because it is relatively easy to let yourself be lazy on the passes prior to your shortest line, because you can get away with it.

 

Why do I always figure this stuff out at the end of the season, write it down, but then can't execute it in the spring!

[Razorskier1]

Than, one more thing. If I focus on the lean being right, and then stay away from the buoy, the finish of the turn seems to take care of itself. It is almost as if I don't do anything other than let the handle out, bring it back in and hang on.

[Than_Bogan]

Sounds like perhaps your goal can be translated to minimizing your maximum speed. (Unfortunately, that's a mathematically unfriendly quantity, but that's MY problem...)

 

Also, I take it as a given that the "optimal" (again meaning lowest effort, purposely leaving "effort" ambiguously defined for now) path only achieves the minimum required amplitude -- i.e. that the rope does not reach any higher angle than it must for the skier to round the buoy. If I can assume that a wider course is harder, then this is clearly true. And the fact that a wider course is harder is "obvious" (the classic phrase used by a mathematician who can't prove something)!

[acmx]

@Razorskier1, that's an interresting description of ur 38 gate. Do u think the boat is pulling u up when u say u let it win? I do what u said u used to do,wail on the bottom, and it doesn't work very well for me.

[Than_Bogan]

Stepping off the Math Train for a moment, I've had an extremely similar experience to Razor on the -38 gate. I tried to document what (little) I know about it here:

 

http://www.ballofspray.com/forum#/discussion/3704/keys-to-my-best-season-ever

 

See the part about the gate.

[acmx]

Than, have u ever read the stuff David Nelson wrote about pendulum theories and math as it relates to skiing? If not I think u would love it!!!!!

[Than_Bogan]

@acmx Yes, I have studied that extensively, and I used it as input into my investigation. He opened my eyes about a few things. But I think ultimately I don't want to comment on his work, but rather just present my own Grand Unified Theory -- someday, if I ever make the time!

[elr]

Than -- Is minimizing accelleration model friendly?

[Than_Bogan]

@elr Sort of. Firstly, I must admit that we* haven't managed to solve out what we want to solve out for ANY metric, but I kinda think we're going to get there eventually.

 

*I have another even mathier guy helping me out when we can find the time, which lately we haven't.

 

But, in principle, any time to want to minimize the maximum of something, it tends to be mathematically harder than minimizing the average (or sum) of something. Average speed is probably not very meaningful. Average acceleration magnitude does have some hope of being meaningful.

 

But it's strangely challenging to even figure out exactly what is meant by acceleration. If we just define it in the "usual physics way" as the magnitude of the (vector) derivative of the (vector) velocity, then it's a little unintuitive, because it's pretty large directly behind the boat and at the apex of the turn. But in each such spot it is perpendicular to the skier's direction. Most people tend to mean "acceleration component in the current direction of travel" when talking about acceleration, because that's when you can feel a big change in your *speed* as opposed to a big change in your *direction*.

 

And THAT definition of acceleration is extra-unfriendly, as it turns out!

[Ed_Johnson]

I try to keep it simple by following the rule, "That for every action, there is an equal and opposite reaction." This tells me if I use one side of the course to accelerate, I need the other side to decelerate. The centerline being the defining point. Anytime I load beyond that point I carry to much speed down course and into the buoy. For me it takes the right balance, especially at shorter line lengths.

[elr]

@Than - I believe I was thinking about it in the "physics" [F=MA] sense, trying to minimize differences in velocity, both acceleration and deceleration. Less angle behind the boat or a higher speed turn would result in less force/acceleration.