Hey Guys,
Just saw this thread and thought I should share a small calculation I had experimented a couple years ago. PLEASE, axcuse my "scientific english" as I studied in french and therefore might lack some vocabulary. Hope you understand it anyways.
As every physical problem I guess you could solve it with the right amount of equations, but as scientists all know that tends to be un-doable in real life given there are so many equations and formulas to take into consideration.
So we tend to work with "Models", which are simplifications of the initial problem.
My model was pretty much the one described by John and others in this thread, I focused on the MAIN issue beeing the distance the skier has to cover AROUND the boat, or "in reference" to the boat, or I don't know how you should say it in English.
Indeed, if you look at it from above, the skier draws the path of a circle (well, a fraction of a circle) around the pylon, so i thought I would just measure the distance the skier has to physically cover, by going from one buoy to the next, at different rope lengths, hoping to find something.
I imagined the skier to be standing straight on his legs (obviously wrong) and with the handle perpendicularly above his bindings all the time (again, wrong in real life), and I also imagined the skier would go to the buoy and immediately change directions to the other buoy, as if there was no inertia no nothing.
And for rope lengths underneath 38 Off, as we all know, the handle does not reach the buoy, so I said the skier had to "swing" around the handle for a distance that could cover the gap, (basically another fraction of a smaller circle, around the handle this time). (Again, very wrong in real life, but if we take these hypothesis and proceed with a consistant calculus, maybe we will see some trends? At least that's what I was exeprimenting at that time)
Well please find attached a screen shot of my Excel file, as you may see, I measured rope length (in M and in ' Off), then the Angle (same as the one described in the picture posted by @Porkfight) in Radians, then the additional distance to actually GET to the buoy when skiing > 38Off, and the distance covered by the skier, in meters (sorry), to physically swing from one given buoy, to the next. (You could then say the slalom consists in 6 times such a swing more or less).
I also measured the average speed "around" the boat, or in reference to the boat, meaning that same distance between two buoys divided by the time it took to get there (boat running 34mph, as this is my case).
I also measured the incremental change to try to understand why some steps in rope change seem harder then others, as you can see not all "steps" are equally difficult.
I hope this makes sense to some of you, and please take into consideration that I just tried this for my personal purpose and do not pretend to hold any greater slalom wisdom or whatsoever, I just thought it fitted the topic ;)
Enjoy.
Romain.