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Constant Acceleration: A mathematical oddity


Than_Bogan
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Warning: 100% useless arcane knowledge ahead.

 

Although I continue to struggle to model what I want and find paths that optimize something worth optimizing, my exploration recently came across something just plain strange:

 

Under some circumstances, it is possible to navigate the course while the magnitude of the acceleration at the rope handle is a constant.

 

A key word here is magnitude. The *direction* of the acceleration is constantly changing, even on this "magic path," but the magnitude of that acceleration is constant throughout.

 

I find it quite surprising that this is possible. And -- as far as I can figure -- it's only possible if the maximum angle that the rope achieves is approximately 0.72 radians, or about 41 degrees. If you assume a skier can extend about 2m from the end of the handle, this is right around what must be achieved for 28 off. (At longer lines it is also possible, if you choose to go wider than required.)

 

Just how useless is this fact? Well, this path isn't even the one that minimizes the maximum acceleration. (That's also a little susprising from a math perspective.) At that same rope length (i.e. required max angle) it's possible to have a varying acceleration magnitude that is always less than the path with the constant acceleration.

 

So this is pure mathematical oddity. But it's pretty odd. (You may have to trust me on that...)

 

Standard disclaimer: It's always possible to F up math, and I can only produce said path by numerical simulation of a differential equation (as opposed to a far more desirable analytical form), so this may not even be true. But I sure think it is.

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I'm willing to accpt the idea, but what about the 'unload' at edge change, the instant just prior the apex of the turn, and the shape or volume of spray generated at different parts of the course? If you pick a given radius for a turn you can calculate (sort of) the g force on the ski and skier- is that the same as the magnitude as the skier takes full down course load into the wakes? How can it be measured?
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Bazinga indeed. I remain confused why so-called normal people watch that show. I guess there's the audience laughing with the nerds and the audience laughing AT the nerds?

 

Dusty -- I hope to eventually make some kind of attempt to answer your question, but as you can imagine this is pretty low priority!

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Than,

I'm not sure I'm considered normal ,but I used to remote past that show real quick. Got bored with 2.5 Men and watched an episode. Got to know the characters and now roll on the floor almost every episode. I laugh for both reasons. Go figure?

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Ok, I decided to be a nerd for a second, I looked at my Differential Equations book. @Than Bogan, are you considering this a free or free damped pendulum, or is it even an pendulum at all. The only way I think you can achieve constant accel is in a frictionless world. I could be very wrong as I sort of despise differential equations.
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Although there are some similarities, I am not using any sort of pendulum model, as that is far too restrictive. A skier can apply force at any time, and in many possible directions, and so has significant control over the path taken. A pendulum, of course, has no control over where it goes.

 

So I am modeling the problem from the ground up, making assumptions only where I must in order to get somewhere in my analysis. (Example: I ignore rope stretch.) Ultimately, I'd like be able to suggest some notion of optimal paths, depending on what one wishes to optimize. I've made "progress" recently but remain very far from anything really worthwhile.

 

If you (or anyone) would like to know a lot more, I suggest sending me an email. I can't even publish equations in this forum, and that's probably a very good thing!

 

nathaniel_bogan AT alum DOT mit DOT edu

 

P.S. All sane people despise differential equations. And non-linear ones are extra nasty!

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@mrpreuss Thanks. I'm tryin' my best!

 

Ironically, I think most skiers are quite aware of the level of control that they have, but when they start trying to do some kind of analysis, they "forget."

 

The other extreme is a bit tempting, too: to assume the skier has TOTAL control of the path. That is, of course, also wrong, because the rope, boat, and of course buoys provide significant constraints.

 

But when you try to model it "right," even with a lot of simplifiying assumptions, this math gets damn hairy damn fast. That's why so far I've only discovered wacky oddities rather than a full ability to optimize paths for specific goals.

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@Dusty To help me answer your question, I made a plot of the direction of the acceleration. Someday I'd like to make a really pretty picture of this that shows vectors on an actual path through the course. But I don't have the time for that just now.

 

Next, some reminders: I am not "measuring" anything. I am *computing* the acceleration *at the handle* by the standard physics method of taking the 2nd derivative of the position in each of x and y. The magnitude of the acceleration is then defined by sqrt(ax*ax+ay*ay). A path is (indirectly) defined by the angle of the rope as a function of time.

 

And everything I'm talking about here is just a POSSIBLE path, not necessarily the one anyone would want to follow.

 

Ok, with those caveats, here's the best I can do for an "answer":

 

At the edge change, the direction of the acceleration is roughly perpendicular to the course and toward the centerline. In other words, the handle is losing outbound velocity, but the component of the velocity in "x" (which I've defined as the boat's direction) is barely changing.

 

The acceleration directly behind the boat (again, on THIS path) is exactly in the direction of travel of the boat. At *this* particular point, this is very much like the pendulum analogy.

 

From the center going outbound, the acceleration begins to rotate toward the centerline, which is semi-intuitive because you begin to lose outbound velocity. Roughly at the edge change, the acceleration points directly inbound. Some time after the edge change, the acceleration rotates all the way to point inbound and slightly backwards, because now you are losing outbound velocity *and* slowing down in the direction of the boat's travel. Then after the apex, the acceleration starts rotating back toward perpendicular.

 

Somewhere around the hookup, the acceleration is actually aligned to the direction of travel! I believe this is true of most "sane" paths and this is the acceleration that I personally "feel" the most -- probably exactly because it is aligned to the direction of travel.

 

Heh, sorry for the wall of text. Sadly, I tried to keep it short...

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