Speed through the course

[eleeski]

What is the minimum and maximum speed a skier hits in the course? How much difference is there from the optimal path to the worst (eleeski) path?

Than, this is right up your alley!

Eric

[Dusty]

Eric- If I can find it... a while back, a skier gave me a table of peak speeds through the course, based on boat speed. It had a detail of the formula used (unknown where it came from... Waterski magazine maybe? anyone?) I will post the formula here if I can find it. I bet Than can make sense of it and prove or disprove its claims. As I recall, there was disclaimer that the formula broke down at/after -38, because of line length= buoy width?

[Gloersen]

The format didn't come through; what appears as a 0 after symbols or 2 digit #'s below = "degrees".

 

Just approximations, down course slide not considered.

 

What about VRope?

 

 

If:

VRope = Rope Vector Velocity

Ω0= Rope angle (to boat)

VBoat = Boat Velocity

 

Then:

VRope = Cosine Ω0 × VBoat

Ω0 = 900, Cosine Ω0 = 0 × 34 mph = VRope = 0 mph (rope speed at the buoy 41 off)

Ω0 = 760, Cosine Ω0 = .242 × 34 mph = VRope = 8 mph (rope speed at the buoy 39.5 off)

Ω0 = 690, Cosine Ω0 = .358 × 34 mph = VRope = 12 mph (rope speed at the buoy 38 off)

Ω0 = 600, Cosine Ω0 = .500 × 34 mph = VRope = 17 mph (rope speed at the buoy 35 off)

Ω0 = 00, Cosine Ω0 = 1.00 × 34 mph = VRope = 34 mph (rope speed behind the boat)

 

What about VMax?

 

Assume:

 

VMax = VHandle = VSki

(for the most part skier speed is equivalent to handle speed)

 

Ψ0= Skier path angle

Ω0= Rope angle (to boat)

 

VHandle =VRope × 1/Cosine Ψ0

or

VHandle = [Cosine Ω0 x VBoat] × 1/Cosine Ψ0

 

Assume Ψ = 530 behind boat (1/Cosine 530 = 1.701)

Then:

Vmax= 34 × 1.701 = 57 mph

 

A 15 off skier probably doesn’t generate a 530 angle behind the boat, but for the sake of illustration if one assumes a skier path angle generated somewhere in that range, think about the ΔV required as the line shortens; 0 to 57 to 0 mph in 2.68 seconds at 41off, 34mph!! For that matter the ΔV’s at 35off are quite impressive!

 

[Wish]

Man I feel dumb.

[thager]

Bazinga! Me too. Was that sarcasm?

[lpskier]

Does that come in English?

[Than_Bogan]

(I'm gonna try to make a shorter post later. Feel free to wait for the "executive summary.")

 

This is indeed right up my alley, but it turns out this problem is Really Freakin' Hard. I've been trying for some time to develop the ability to compute optimal paths based on a given criteria and the math gets real ugly, real fast. The best case is usually a nasty differential equation that can only be "solved" by numeric simulation, and the worse case is so ugly that we can't even get it down to a differential equation.

 

On the other hand, the problem of determining the maximum and minimum speeds, at least to a good approximation, of a given path, is pretty doable.

 

One surprise (although now that I see it, it feels obvious) is that the maximum speed generally does not occur directly behind the boat, especially at short lines. It actually occurs a bit after that, and the reason is that the skier is now still going fast in the "accross" direction, but is *also* "catching up to the boat" in the "forward direction." So the net speed continues to increase on the other side of the wake -- you can almost think of it as the swing of the rope is pulling the skier ahead faster. Naturally, this doesn't continue very long, so the peak speed is generally just outside the wakes.

 

Similarly, the minimum speed doesn't occur out at the apex, but slightly after that, and for the same reason: The "swing" of the rope is now in the backward direction, so it continues to reduce your speed for a few moments until the whole acceleration patterns begins anew.

 

What I can do is report the min and max of the rope handle (which is a pretty good approximation of the skier's minimum and maximum since they happen to occur when the skier's c.o.m. is fairly near the rope handle) for a variety of rope lengths, and assuming a path that I have reason to believe is *near* to optimal, although I can't yet prove it. I'll try to put this together when I can, but I'm really not too sure when that will be.

 

As far as the suboptimal paths, there is no obvious bound. If you make things bad enough for yourself, you may have to achieve Ludicrous Speed if you want to actually make it. So the worst case is limited by the speed you CAN achieve, not the speed needed to complete the slalom course efficiently.

[Roger]

The highest measured speeds I've seen are 57mph at 58k/10.75 (Andy Mapple) and 47mph at 55k/11.25 (Deena Mapple). The pro tour comentators used to speak of 70mph speeds through the wakes, but I never believed that even before Dave Benzel started measuring speeds.

[eleeski]

Just to make things difficult, wouldn't the magnitude and timing of any variations in boat speed have a huge magnifying effect? Leading to the question: what is the best ZO setting?

Eric

[Gloersen]

It is very complex as Than points out. However the basic tenet that ΔV increases as the line shortens is perhaps good to conceptualize as a training aid, then again perhaps not; depends on one’s analytical nature. The spot crossing the course we VMax is probably of the essence.

 

The skiers VMax is inextricably linked to the velocity of the rope along its vector and the tangential vector along which the handle travels at any given point. So VMax can certainly be at the 2nd wake if good technique is executed (handle control); if the tangential vector of the handle path to that of the rope is at an angle greater than when directly behind the boat such that the VHandle = [Cosine (rope angle) x VBoat] × 1/Cosine (ski direction angle)] product is relatively greater.

 

Due to poor technique I’m quite certain that as the line shortens my VMax is before the 1st wake and my (ski angle direction) diminishes approaching the 2nd spray = bad = something upon which to focus for correction this Spring. Rather than “locking & loading” (static technique); creating the “velocity swing” (dynamic technique through the spray to spray work zone).

 

[Than_Bogan]

Eric: "Just to make things difficult, wouldn't the magnitude and timing of any variations in boat speed have a huge magnifying effect? Leading to the question: what is the best ZO setting?"

 

Well, maybe. My first instinct is that the real-world variations of the boat actually have a very slight *dampening* effect. I strongly suspect that the boat's minimum speed is happening right around the same time as the skier's maximum, and vice versa. This is because the maximum load on the boat occurs directly behind the boat (true of nearly any "sane" path as far as I can tell), so unless ZO over-compensates for that load (which I think skiers would absolutely hate), that's going to be about where the boat speed is slowest.

 

Of course, I've noted that the skier's maximum speed occurs a little after that. So if ZO were set to "gun it" at that particular point, it is conceivable that ZO could slightly increase the skier's maximum speed.

 

But, even if that's the case, it doesn't necessarily tell us much! Speed isn't necessarily a bad thing! In fact, a long time ago I discarded "minimize the maximum speed" as a good criteria for an optimal path, because a lot of that speed just comes and goes "for free" due the geometry. What's important is the effort that the skier has to put in, and sometimes carrying more speed at certain spots ultimately leads to less effort.

 

I really would like to be able to model details like boat speed variation and comment "intelligently" about such details, but the math is border-line intractible even with a lot of simplifying assumptions. So I'm not optimistic.

[PT Mike]

Didn't WSM do an artical back in the day w/ Wade Cox & a radar gun? Pretty sure he was clocked at just under 60 MPH crossing the wakes. A mere mortal running 35 was 10 MPH slower. He said that's why a lot of guys run 35 but not 38 The game sure has changed, hasn't it?

[eleeski]

Minimum speed would be just a couple of triangles hypotenuses added to the distance. The minimum path is just a bit longer than the boat's path so the speed will be just a bit higher. But that is not a realistic path so I may not have posed the problem well.

 

Gloerson's analysis based on angle out of the buoy is interesting but the turn angle is way too variable (at least if I'm skiing). Maybe speeds do react primarily to angle. The overhead view of a top skier in the course would be enlightening.

 

The radar guns of Andy and Deena are suspect on two fronts. One, does a radar gun get an instantaneous speed or a dampened out average? Two, Andy and Deena are smooth skiers. I wonder what happens when they scramble? I would suspect that the speed swings are much higher when not following the smoothest passes or skiers.

 

I know I personally have come ripping into the buoy at warp speeds (time slows down giving me time for my life to flash before my eyes). I won't comment on whether I made the next buoy.

 

The feel from the boat does have significant effect on how I ski. Maybe the magnification of the skier speed is not where the boat speed variability manifests itself. But the nature of the boat's speed variations do matter to my skiing.

 

Than's claim that maximum speed occurs at the second wake has ramifications on training technique. I'm intrigued!

 

Eric

[Than_Bogan]

Eric: "But the nature of the boat's speed variations do matter to my skiing."

 

Certainly true! Hard to quantify.

 

Eric: "Than's claim that maximum speed occurs at the second wake has ramifications on training technique. I'm intrigued!"

 

Hm, it does? I think this is just more a consequence of the geometry. Even if you are already starting to ease up your pull, the fact that the rope changes from swinging you opposite to the direction of the boat's travel to the same direction as the boat's travel causes your net speed to increase. Theoretically you could force the maximum speed to occur in a very different spot, but I believe any such path would be extremely suboptimal.

 

Btw, it's actually beyond the second wake at extremely short lines on most "sane" paths that I've considered.

[Roger]

@eleeski - "The radar guns of Andy and Deena are suspect on two fronts. One, does a radar gun get an instantaneous speed or a dampened out average?"

 

Not done with radar, done with LISA (I believe that was the name anyway, a device for measuring a number of skier parameters).

[rq0013]

A ski friend of mine had a radar gun at the malibu open this year. Whitney McClintock had the highest speed of the day, 47mph. Tgas ran 44mph at each rope length no matter what. Will Asher was up and down with speeds around 41mph. He missed nates set, which would have been interesting to see.

 

 

[Roger]

The 36mph skiers were skiing slower than Whitney; doesn't sound right...

[teammalibu]

Me think radar gun broken! Need to equip smokey with that model!

[SkiJay]

Interesting question. Are we talking the speed of the skier's head, hips, feet or the handle?

I bagged a GPS and skied with it under my vest. It said my chest clocked a 57mph top speed. Oddly enough, it showed my lowest speed in the course to be zero!

[rq0013]

I dont know, thats what they got.

[Dusty]

teammalibu- Quite likely... Radar guns will measure the minimum speed of an object. They are subject to an algorithm, to 'see' the closest, largest or fastest in a 12 degree "beam" (About 20 feet across at 100 feet distance). Anything moving around in the beam can get measured- like spray? A "Lidar" unit measures what is is aimed at (think crosshairs). Would need a reflective target- like a square of aluminum foil taped to a skier vest? But it is very accurate and would yield some precise, empirical speed measurements.

[MichaelGoodman]

With a perfect skier pass the skier has to travel about 20% farther than the boat. At 36mph the skier would have to avg

over 43 mph the top speeds would have to be much higher than 43 even with a pefect skier path.

[ktm300]

"Maximum load comes right behind the boat" I always assumed this too. However, I rode in the boat with one of the best and smoothest 55k guys out there with a strain gauge on the rope connected to a laptop with software that would plot the load through the course. Max load is about one ski length inside the buoy line. Watch the West Coast slalom video. Same thing; max load almost right off the ball. Until I witnessed this, I was truly confused by all of the "light on the line" talk. 650 lbs of load may be lighter than 1000 but, it isn't light.

 

[colo_skier]

ktm300 makes a good point. Does max load equate to when maximum acceleration occurs? If that is so is our intent to get as far back to the wakes as possible before loading the boat?

 

[ktm300]

colo skier That is kind of what I was taught by a great coach. Got a few sets coaching with Jeff Rodgers and he said "you are almost to the first wake before you are picking up the line". He wanted me to pick up the load from way out. As I had spent a lot of time and effort to learn the "ride the ski back to the wakes and pick up the line late" method, I just hit delete. However, Jeff can run some serious buoys. From the boat it is a thrill just to watch it. I know for certain that he is much stronger physically than I'll ever be. It isn't weight lifter strong but tendon and ligament wiry bio mechanically strong.

[Phil2360]

So on a tangent, who's willing to take their Smart Phone Skiing.

We plan to.

Plenty of apps out there that do acceleration logging.

Just need to find a nice compact waterproof case to put it in under your vest.

Hopefully one that floats.

 

Other thing that would be fun to jump for is one of these:

 

http://www.gcdataconcepts.com/products.html

 

 

[Bulldog]

Look up the "Life Proof" case for Iphone 4 they claim it is totally waterproof. A couple of guys at work have it (Local Fire Department) and they swear by it, drop proof and dust proof as well.