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# Question for PhysicsMan

PM,
I was watching a tape of a slalom race the other day, and it reminded me of something I've been wondering about. In the commentary, Todd Brooker claims that rapid-gates (in SL) reach speeds of 600 mph at their tips when they're snapped down to the snow. I don't know how to calculate the probable speed, based on the speed of the skier displacing the gate, because I don't know how to take into account the differing speeds of the different parts of gate--they don't simply swing in a straight line which would make the calculation easy. How do you take into account the whip-like action of the tip of the gate, against the much lower speed of the base which doesn't behave that way?
Unfortunately, I don't know any simple way to perform the calculation that you asked about. Here’s why. A computationally very similar problem is to calculate the tip speed of a bullwhip. Believe it or not, after decades (maybe centuries) of interest, a paper is just coming out in Physica D which finally does this accurately:

http://www.math.arizona.edu/~goriely...s/whipwave.pdf

If you take a look, you can see that their calculation certainly doesn’t involve particularly “deep” physics, but it is involved and tedious, and much attention to detail was needed to get their model to behave properly. Exactly as in the case of the whip, as you pointed out, rapid gates are flexible, different parts have different physical properties and boundary conditions, and this presumably causes the velocity to vary dramatically as you go along its length. This is why a simple analytical estimate isn’t accurate, and a numerical simulation has to be used.

OTOH, I think that the most direct way to answer your question is not by computations, but by experimental observations. For example, one approach might be to look for still photos of a rapid gate taken from the side, just after a racer hit it. If the shutter speed that was used is such that the racer appears just a bit blurred, look at the tip of the gate. If it is more blurred than the racer, it is moving faster than the racer. The ratio of the blur size of the racer to the blur size of the tip of the gate will give you an approximate idea of the speed of the gate (assuming some nominal value for the speed of the racer). Obviously, there are possible complicating factors (eg, the weird motion blur patterns of focal plane shutters), but this approximate approach does not even require that you go out and take a picture yourself, just find an appropriate one and look at it.

On the other hand, if you are willing to go out and take a photo yourself so that the shutter speed, shutter/camera type, illumination type, and angles are all under your control, you will be able to come up with a much more accurate estimate of the tip speed of the gate using the "blur" approach.

Now, in the realm of total seat-of-the-pants guesswork, my gut feeling is that while 600 mph is not at all high for a bullwhip, 600 mph IS a very high number for a rapid gate. The reason I think this is that it takes substantial curvature along the length of a whip to get it to “crack” (look at the illustrations in the above article), and I don’t think the material of construction of the rapid gate will allow this much curvature without breaking. In the limit of no curvature, the gate will act like a simple rigid rod pivoting around one end and hit somewhere above the pivot point. In this case, you might at best get a factor of 2 to 5 multiplication of the speed of the skier, but that only puts the tip speed in the 40 – 150 mph range, not at sonic velocities (assuming the skier is moving at 20 – 30 mph).

I wish I could give a more definitive and simple answer, but some problems just don’t lend themselves to this (as you pointed out).

Interesting question.

Tom / PM

[ November 30, 2003, 11:44 PM: Message edited by: PhysicsMan ]
I had heard that speed to be about 200 MPH. I know that Volkl had a lot of problems with their skis delaminating because of it a few years ago.
I also have seen sequences where the gate bends into an arc and the tip spears the racer in the face. In the 89 World Championships at Vail a German racer actually got "clothslined"(head snapped back and feet out from under her) by this process.
I had also heard 200mph. I think 600mph is a little excessive. Do you think there would be a way to directly observe the speed rather than calculate it?
Thanks, Physics Man. I never thought of using the 'photo method' you described, but I don't see why it wouldn't work. I had done the same sort of 'rigid rod pivot' rough calculation you described, and would guess that the speed of the gate at its tip would be significantly higher--the gates do bend and load up a fair amount, although not nearly as much as the end of a whip does, of course.
In photos that have any blurring of the racer the end of the gate is usually so blurred as to be completely indistinct (if it is already on its way down toward the snow). I may give the photo thing a try.
I had just assumed that there would be way to do the calculation that involved math/physics that is beyond me.
Oh yeah, I only heard the 600 mph figure in one race's commentary (thanks, Todd Brooker), which is why it struck me. I had often wondered how quickly the end of the gates moved, and that figure did sound quite high.
Even at 600mph you're still in reasonable acoustic regimes. Put a cheap speed-to-pitch whistle on top of each gate?
Doesn't the gate "crack" just because it slams into the ice/snow/skier/whatever? When a hockey player knocks his stick on the ice - with all due respect, I can't believe that the hockey stick slams into ice at 600 mph.
Quote:
 Originally posted by comprex:Even at 600mph you're still in reasonable acoustic regimes. Put a cheap speed-to-pitch whistle on top of each gate?
Sure, why not? A racer coming down the course would sound like my daughter's Guinea Pigs when they are about to be fed: ...wheep! ... wheep!

Interesting thought, but unfortunately, 600 mph is getting close enough to sonic that the flow is quite compressible, so a calibration of the whistle would have to be done in the 200 - 600 mph range (ie, you would need a wind tunnel - grin). Corrections would have to be made for the huge chirp resulting from Doppler because the source is moving (yeah, you could try for 90 deg, but good luck), a good data acquisition and analysis system used since the duration of each whistle would be so short you couldn't use your ears, etc. I think analysis of a blurry photo would be better and easier.

If you want to do an even better (but still cheap) job, borrow an old HP variable frequency strobe light from a local university, position it across the course from the camera with the gate between the two, and set the strobe light for 100 usec spacing between the bursts of light. The image of the gate will appear as a series of shadows like the early (famous) Harold Edgerton (sp?) photos. Measure the distance between them, and you have a very accurate number for the speed.

...wheep! ... wheep! I'm outta here.

Tom / PM
Quote:
 Originally posted by AlexG:Doesn't the gate "crack" just because it slams into the ice/snow/skier/whatever? When a hockey player knocks his stick on the ice - with all due respect, I can't believe that the hockey stick slams into ice at 600 mph.
You are absolutely right. I have no idea how someone came up with the 600 mph number, but I certainly hope they didn't use the "I hear a clap" reasoning that you correctly argued against.

Actually, even the lower 200 mph number sounds high to me.

Tom / PM

[ December 03, 2003, 02:16 PM: Message edited by: PhysicsMan ]
What you aren't taking into account is that Todd is Canadian...that means 600 mph Canadian, which, like their peso, has to be substantially reduced.

Its true with all Canadian measurements.

[ December 03, 2003, 02:24 PM: Message edited by: Stu Pidasso ]
Well written paper. Thanks for the cite.
Could the gate be timed..from the moment of impact until the tip hits the snow and then the distance traveled measured, to figure out the speed.
That would only give you the average speed, which would be substantially lower than the top speed achieved by the tip of the gate.
Stu –
------------

Lal - Good to see you! It’s been a long time since I last spotted you on Epic. This is a great place with lots of nice people and good, on-topic discussions. Why don't you take off your coat and stay for a while this time?

For those of you that don’t know him, Lal is a very experienced and knowledgeable skier who could contribute much to the discussions on Epic. Father of a racer and well-known retro-grouch that he is ( - hope you don’t mind the ribbing), he does have this somewhat odd penchant for long skis, but other than that, he's a great guy.
------------

Lucky – Pete’s answer was right on the mark.

Tom / PM

[ December 04, 2003, 01:45 AM: Message edited by: PhysicsMan ]
Please pardon me for changing the subject, but given the many variables involved, do heavier/more powerful racers achieve faster times than lighter/smaller framed racers on less steep race courses due to their gravity/momentum?
Yours is one of the all-time classic, non-trivial "Physics of Skiing" questions. The first time I ever saw it discussed on the internet was in 1995. For your amusement, take a look at the first post in the thread http://tinyurl.com/y36f and browse forward through the entire 188 posts. Unfortunately, this RSA thread contains a lot of misconceptions, errors in elementary physics, as well as out-and-out silliness, so its not particularly useful, except in terms of providing examples of the the type of errors and misconceptions that can be made in such a discussion. Similar threads regularly appear in many ski forums.

I believe there was a fairly good thread on this subject here on Epic a year or two ago, but with the poorer search engine of Epic compared to Google, I couldn't lay my hands on it at the moment. As I recall, one writer in the EpicSki thread took the emperical approach and simply looked at the undebateable historical evidence to see whether heavier or lighter racers tended to win in the various events (ie, DH, SG, GS, SL). I think I remember the conclusions, but hopefully someone will be able to provide the URL of that thread and the real numbers.

Even the seemingly simple theoretical situation of sending two weighted, riderless, identical skis down a hill has substantial complications. In a vacuum, and with no friction from the snow, they clearly would both arrive at the bottom of the hill at exactly the same time, no matter how steep the hill, or how much weight each was carrying. On the other hand, in the real world, the arrival times will depend on the weight they carry because the several forms of friction acting on them depend on weight in quite different ways. Some are independent of weight (eg, aerodynamic drag), some increase with weight, and some decrease with weight, everything else (slope angle, ski and base prep, etc.) held constant.

For example, on a shallow slope covered with deep powder, the ski with the heavier weight will indent the snow more and might even get completely bogged down, whereas a ski carrying a very light weight would tend to ride on top of the snow and have not only less total friction, but also less relative friction (ie, frictional force divided by the skier's weight - like a psuedo coefficient of friction).

Another situation: On cold, packed, abrasive snow, the extra weight carried by the heavier of the two skiers skier might provide enough down-the-hill force to break his skis free (ie, overcome the "stiction"), whereas the lighter skier on the same slope might only have barely enough down-the-hill force to break free, and so will accelerate more slowly and have a lower terminal velocity.

As a final example, at very high speeds, wind resistance is often the dominant factor, so, assuming the heavier skier and lighter skier have approximately the same frontal area in a tuck, their aerodynamic friction would be the same, so the heavier skier will reach a higher terminal velocity.

Real skiing situations are more complex than the above "ideal" cases, and also involve the muscle power and skill of the skier, so the real answer to your question must always be empirical, from the historical record (as mentioned above). I'm pretty sure I remember the "answer", but hopefully, someone will post the link or equivalent data.

HTH,

Tom / PM

[ December 08, 2003, 03:53 AM: Message edited by: PhysicsMan ]
Thanks for allowing me to revisit those discussions.The complexity and science of getting it right at race level is daunting.It makes me think that course setting must be quite an artform and carries considerable responsibility as well.
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