Slalom Ski Path Physics (Matlab Model)

[lundberg]

I was reading the "Does leaning harder make you go faster across course?" tread and there was a question "are you going slower than the boat on your way to the wakes" and it got me thinking.......so I tried to model it in Matlab.  I parameterized it as much as I could (boat speed, rope length, reach, swing velocity profile) with a few assumptions (no slack and not skiing any wider than you need to).   The swing velocity profile drives the path and dynamics.  Here is what I looks like.  A couple of things stood out to me.

1) The shorter the rope the faster....and slower you go

2) Your max speed over the water is much later than your peak swing speed

3) If you are pulling all the way to the buoy you are going really fast

4) Yes you go slower than the boat, sometimes way slower

 

What do you notice?

 

Example: Max swing speed at center line

 Example: Max swing speed before centerline

Example: Max Swing Speed after Centerline

 

@AdamCord I think this lines up with "GUT" (https://denaliskis.com/pages/gut-104) except that a minimum swing speed does not make a straight line handle path unless the rope is infinitely long.  

Edited February 19, 2023 by lundberg

[Taynton]

insert homer drooling gif here...

man if you could track your path accurately and compare how amazing would that be?

 

[MDB1056]

IMO- way too much information. Please crystallize into 5 lines or less 

[lundberg]

@Taynton If you had rope angle over time, rope length and boat speed you could get the same data.  Seems like you could add that to Perfectpass or Zero-Off and know quite a bit.

 

@MDB1056 I agree!  How about this,

15off 34.2mph running an early line, speed off ball ~28mph, 4m past centerline ~50mph and will see 1.5gs

38off 34.2mph running an early line, speed off the ball ~19mph, 5m past centerline ~57mph and will see 2.5gs

Pro running 41off 36mph running an early line, speed off the ball  ~15mph, ~6m past centerline ~63mph and will see 3.4gs

 

[ScarletArrow]

[insert chevy chase gif]

“I was told there would be no math.”

[lundberg]

@Drago actually you do. As you gain on the boat from centerline to apex you go faster than the boat. As you go from apex to centerline you lose ground to the boat (it pulls ahead) and you go slower. The really interesting thing is how much slower you go when the line is short. For the line to stay tight you must slow way down. 

[S1Pitts]

3 hours ago, lundberg said:

What do you notice?

it seems between between comparable graphs, there is not a terribly huge difference @ 34 & 36mph.....but that could be my blooodshot eyes that have been awake for almost 20hrs.

[lundberg]

@S1Pitts Handle paths are the same (driven by the swing speed profile. 

Here is the difference in speed at 38off for 34.2 and 36.

Edited February 19, 2023 by lundberg

[503Kento]

@Taynton the guys in this video seem to be working on what you want.  Other sports e.g. motocross have been using this type of technology for a while  

 

 

[MDB1056]

Very impressive video. Great to see JB Faisy out there still skiing extremely well 

Edited February 23, 2023 by MDB1056

[aupatking]

A WHOLE LOT of info there. Am I reading it correctly that the speed AT THE BALL (or maybe better said, “at the turn”) is about the same for all line lengths shown? 

Edited February 22, 2023 by aupatking

[lundberg]

@aupatkingyes and it should equal the boat speed. With the assumption that the rope stays tight, the handle swing will go up on the boat, slow as it approaches max width / height and when it changes direction back towards the wakes it should match boat speed.  As I type this I'm wondering if the difference in ski speed and handle speed is the highest at this point since it is moving from outside the handle line quickly to in line with the handle. (Looks like I'll need to update the model 😉)

[aupatking]

So carrying speed through the ball would seem to be non existent. So the “fastest ski” would be a ski that accelerated after the ball quickest, not one the “carries speed through the turn”?

[RomanMia]

Very cool

[MarcusBrown]

@lundberg  @Drago Intriguing comment about the apex/finish of turn: "The really interesting thing is how much slower you go when the line is short. For the line to stay tight you must slow way down."

Didn't someone once say, that to keep the rope tight and the carved radius tighter, one might need to learn to Turn from the Tail of the ski? - AKA the part of the ski that creates the most drag... 🤔

....or else, SLACK

[MDB1056]

sounds a bit like we're back to almost replaying stop/start skiing of how many years ago?  

[lundberg]

@MarcusBrown

@MDB1056

It is hard to argue with the handle path if the goal is no slack. That being said, the "dynamics" of the swing can radically change the effective path, but what the ski does in relationship to the handle especially around the apex and finish of the turn is not modeled.  It seems like there is a balance between creating more time / margin and minimizing acceleration. "Big" changes in speed is where things go wrong. 

[VONMAN]

Math! Why did it have to be Math! I hate Math!

[chris55]

Would you collect the same informations if you just put a GPS on a skier and another one on the ski ? And then no math for Vonman 😄 just speed and distances

[GSchmid]

Great work Lundberg!  thanks for sharing this.

A couple of observations I was surprised by:

1) peak velocity occurs much later vs. centerline than I would have guessed.

2) skiing an "earlier line" looks like a disadvantage in terms of both load profile on the boat and the change in velocity experienced at the buoy. (is that a correct interpretation)

Not a surprise, but the "late line" simulation looks scary as hell.  That's some crazy speed!  A big penalty for pulling long.

Any thoughts on how this model plays with speed control and where/how the boat should be picking you up?  Seems like an earlier hit (C) would be an advantage because of the asymmetry of the velocity change (ie. early is much better than late).  I'm not sure about the 1,2,3...  

[lundberg]

@GSchmid

2 hours ago, GSchmid said:

1) peak velocity occurs much later vs. centerline than I would have guessed.

 yep, it is because the speed you made "swinging" goes from cross course to down course and they add together. This is how you catch up to the boat. 

2 hours ago, GSchmid said:

2) skiing an "earlier line" looks like a disadvantage in terms of both load profile on the boat and the change in velocity experienced at the buoy. (is that a correct interpretation)

I think so but I also think you end up with less time / width approaching the ball. I'm guessing if you can handle the load, more margin for error might be more valuable. 

As for the speed control, I haven't thought about it nor do I have enough experience to comment. 

Right now I'm wrestling with how to model the ski location with respect to the handle. I think there is some magic with how your ski carries speed while the handle swing comes to a stop and goes back the other direction. 

[DW]

@lundberg - handle to ski relationship at the ball may be another pendulum, perhaps the arm extension / retraction could be kind of a spring.  Now simply superimpose them together!! 🤣

We've used Dartfish in motorsports for a long time to analyze & compare things like racing lines, etc.

@503Kento - once we start down the data acquisition rabbit hole, the sport will get much more expensive for the highly competitive segment of the sport.   Once a few start the rest that want to stay on the same level won't be able to do without.

Edited February 26, 2023 by DW

[503Kento]

@DWi agree it’s yet another money pit for skiers but who doesn’t love a good money pit 😜

if one could superimpose real skier data with some sort of animation of ideal skier data, that would be a super cool coaching aid. 
 

I also think it would be cool to have these stuck on the pros at tournaments. @TonyLightfootand crew could then do some telestration type work comparing different skiers.

[03RLXi]

I'd love to see these for 46 or 49kph and 18.25m i.e. the learning the course speeds. It'd give some insight why the initial learning the course phase is so dam hard and why 3kph speed increase seems to change everything 

[lundberg]

@03RLXi

Here you go.  Interestingly the handle path is the same for all speeds (assuming the same swing profile, max swing speed is just before the wakes).  The big difference is you have to go a lot faster since the time is getting shorter between the buoys.

It would be really interesting to get swing profile data from real skiing to see how it matches.  In a couple months when my boat is out of storage I've got some ideas I want to try.

 

[03RLXi]

Wow, that's awesome. Thank you. I need to take a closer look and think about this. It shows so much. Fantastic!

[03RLXi]

I worked out that the ratio of boat speed to directional speed on 18.25m rope is roughly 1.47x

So I can finally tell my friends I reach approx 1.5x boat speed when slalom skiing.

Once rope starts to get shortened the ratio increases and peaks at around 2x