Scientific help needed!

A snowHead

This is a topic that's just come up on snowRacers, ref. the considerable advantage lighter athletes have in the ski jumping disciplines. FIS has now introduced minimum weights...

I'm trying to work out a 'simple' formula from a study that I found on the Net that can be applied to the advantage extra weight gives skiers in the alpine speed disciplines... Any scientists/mathematicians out there that can help me out?

I've read and re-read this (Croatian) study, but I've got to admit defeat, it's beyond me.

Basically I want to be able to work out - easily - in percentage time terms - the advantage a 50kg racer has over a 35kg racer, a 70kg skier over a 60kg, etc, etc.

Help... Ian, Laundryman, anyone else?!

A snowHead

I'm pretty sure this has come up on www.epicski.com, PG, but I wonder which keywords would unearth the magic thread(s)!

Well, the person's real but it's just a made up name, see?

PG: I've read the paper and from a mathematical modelling point of view it looks pretty ok.

It should be possible to put something simpler together to allow one to measure the advantage in weights but over a simple course and by considering 2-d instead of 3-d. I'll see what I can knock up in a spreadsheet if you're really that interested....and the maths will be a little easier to follow.

Note that figure 2 show's the % change in time versus weight for the courses they've simulated. This chart should give you in approximate terms what you want since all things considered, this is a simulation and the course undulations, small slips, leaning and jumping will have effects that probably outweight this model.

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The model (like any other) is simplified. I don't see reference to the coefficient of friction being altered by the resultant forces as well as speed and temperature. I suppose they could be negligible, or I could be talking out of my @rse. Other relevant variables could be added strength, altered body weight distribution, certainly height.

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biggles76, that would be really useful - thanks. Appreciate that other factors can, depending on conditions, outweigh the effect of drag/weight ratio. Still as you say, over a simple, softish, and in particular a flattish course, its influence can be considerable.

Strength, technique, type of wax used etc, all come into play but from following the same groups of racers around race circuits over a period of years, it became pretty clear that the lighter racers had a considerably better chance of making these pay in steep, technical, icy conditions. That said, you would have to assume that the heavier racer with identical strength/skill/balance would still come out on top in the latter conditions.

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Surely lift is important too and relates to wind resistence, which relates to build (tall and thin (but with nice spready suit v short and plump). So do they have to factor that too, or is it much less important? Does the length of skis currently allowed (which also effects lift) relate to height or weight or both?

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How is this study useful? Is anyone suggesting a handicap system? Are we going to encourage Sumo wrestlers' diet for young athletes?

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PG, Slovenian. [/pedant]

snowball, yes, tall and skinny is definitely not an advantage. Longer skis, if the racer can handle them, can help recover some of the difference, but you'd assume that a heavier/taller racer with equal technique would be able to handle longer skis still.
MartinH, it's useful from the point of view of children's racing where I am involved. Some parents of smaller kids can be disilllusioned by the apparent gulf in ability which in fact is often recovered in later years. Children race in 2 year age groups - ie there can be a full two year gap between two racers, and given the different rates at which children develop physically, I've seen 30kg racers up against 55kg children. That's a very large differential. Flattens out as they get older of course, and then it's "horses for courses", as they slot into the disciplines to which they are most suited.

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In some dinghy sailing, competitors can add weight (up to defined limits) by means of a special waistcoat. Now PG how about using, say, Hannah to do an experiment. Add weight by means of a small backpack. Shouldn't affect drag much and if she's only doing a moderately steep schuss won't affect style or technique or risk injury. She's about 40kgs (I think you said somewhere) so let's see if a 10% addition makes a measurable effect. To eliminate wind effects, etc., you might have to average over a number of runs. If Figure 2 in the study is right, you should get around 1/2% reduction in the time - could be a bit tricky to measure unless you've got a nice long piste to play with.

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kuwait_ian, If I was sailing a dinghy the last thing you'd get me to wear would be a weighted waistcoat. On the other hand what a motivating factor not to capsize!!!!!

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PG, I see what you mean. Do you think it would make more sense for children to be racing in weight groups, like boxers?

MartinH, No definitely not - racing (for children) should come second to having fun, getting in as much time on the snow as possible, free skiing - that's what pays dividends when they get older, not how many gates they can fit in in a day's skiing.

There's a big drop-out rate as competition gets tougher the older you get. Results prove that the clubs that concentrate on fun in the early days may have average children's results, but in the junior/senior categories - when it counts - they climb to the top of the tree.

Scroll down past the French blurb here...
http://scla.alpesprovence.net/cv.htm

And on last year's 'league table' of French clubs, Les Arcs (1st) and Grand Bornand (3rd) have caused a rethink in French race skiing, as pioneers of this approach...

But parents and children racers at least need to understand the physics of what's going on - and the point of this is to give the lightweights a little hope, and make the heavyweights less cocky!

You know it makes sense.

From a quick skim, I think the paper leaves out a number of factors that may make a difference. It all seems to be about air resistance, with nothing on the contact between the ski and the slope. If going straight downhill, then a heavier skier will compact the snow, reducing the coefficient of friction between the snow and ski, and thus will go faster. This might not make much difference on a slick race-prepared piste. Conversely the 'reaction' through the surface will increase with the weight of the skier, tending to slow down a heavier skier. This will have a greater effect the shallower the slope, because it is the component of the skier's weight acting perpendicular to the ski. That's zero when you're plummeting down a 90 deg. slope! Mathematically, the frictional force F is given by F = µ.R where µ is the coefficient of friction and R the reaction. So µ is less for a heavier skier but R greater. R is given by m.g sin ø, where m is the skier's mass, g the acceleration due to gravity (m.g = weight) and ø is the angle of the slope. The factor sin ø is 1 on the level and 0 on a cliff.

Now the tricky bit, which is to do with turning. We need centripetal force to turn a corner. This is provided by setting the ski on edge, which means that the reaction mentioned above now has a component acting towards the centre of the turn. That's, as we've seen, proportional to the skiers weight, but will have no overall effect (to a first approximation) because a heavier skier needs more force (exactly in proportion to his mass) to turn a circle of a given radius at a given speed. (Just as an aside, we separate at the hips to bring our centre of gravity back over the skis, whilst maintaining the edge, in order to maximise this reaction). So far, no difference then. But, I believe it's easier for a heavier skier to maintain a given edge. A component of his weight is acting through the edge of the ski to get it to bite, as well as through the surface of the ski to cause the reaction which provides the centripetal force. The lighter skier who can't maintain the same edge will either lose it altogether, or have to accept a less severe edge, which will mean he'll either go slower, or on a longer radius turn, or (more likely) some combination of the two. I'll show the maths of that later, if anyone's interested.

Set against the above effect, I have an instinctive feel that a lighter skier will get from edge to edge quicker, based on the need of a heavier skier to apply more energy to move his mass laterally and the increased air resistance he will encounter in the lateral movement. If that's right, then it would follow that the optimum weight for slalom would be less than for GS and Super-G. Any info PG?

In conclusion, I reckon that the situation is more complex than the Croatian paper would suggest. That's probably not what you wanted to hear Confused

Otherwise you'll just go on seeing the one name:

Just quickly, a comment on the first para. With you through to "slick race-prepared piste". But then if I read you correctly, you are saying that the shallower slope will have a greater slowing down effect on the heavier skier? If that is the case, it doesn't manifest itself in practice - lightweight racers lose more time on the flatter slopes, assuming a simple course is set. In fact, the differential is increased in 'sticky' snow. How might that be explained?

A snowHead

PG, Taking the 'sticky' bit first, I think that's consistent with what I was trying to say. The weight of a heavier skier will reduce the stickiness under the ski. He might also smash loose stuff out of the way that would slow down a lighter person.

As for the advantage on a flat-ish slope, perhaps it's all down to the start: the heavier, and probably stronger, skier can generate more speed more quickly by skating at the start, which is only bled away slowly (and which he can perhaps top up now and then). Just a theory. But I think it all goes to show that the Croatian paper addresses in detail a small part of the problem while leaving out altogether potentially important factors.

A snowHead

I also think a heavier skier may be able to come down a ski jump faster with a higher speed because he/she has a bigger mass and a larger accelerating force (by force=mass x acceleration with the acceleration fixed by gravitational pull).

However, in the speed that we are talking about, over 120m, aerodynamic force will be dominant. There a lighter skier may be able to glide over a longer distance as with smaller frontal area, on account being smaller in size, he/she should have a smaller lift to to remain air borne.

The trouble is a good balance is needed for a skier to launch sufficiently fast to shoot off the ski jump with a sufficient high inital speed. Thereafter the lighter weight shall help the airborne duration.

Well, the person's real but it's just a made up name, see?

saikee, Although PG alluded to ski jumping, I think he's actually interested in the physics of racing.

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On ice, straight down hill (any degree of slope), yes.

Soften up the surface, and two opposite (but not necessarily equal) effects come in to play. The heavier person will cause the ski/snow interface to be more slippery, to his advantage, but at the same time the reaction through the surface will tend to be greater, to his disadvantage. I would guess the balance to be more in favour of the heavy skier the softer or more cut up the surface.

Put in some hard turns, especially on a hard surface, and the heavy skier will get an advantage due to the extra 'bite' he can get on his edges. Lots of little turns, and the balance may be with the lighter skier, because he can move from one set of edges to the other more quickly.

A push start clearly favours the heavier (really the stronger) skier and will have the most impact on a shallow slope.

However, ask me to develop a set of equations for an arbitrary course on an arbitrary day to construct a fair handicapping system and I'd have a nervous breakdown! (Would make a great topic for PhD research though -- especially when it comes to testing the theory!)

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PG, (and others) Talking about "lift" I thought people would understand I was refering to the original post about ski jumping. Lift is absolutely crucial and is effected substantially by luck with wind. I should have thought a tall person, would be at an advantage, not a disadvantage over a dumpy person of the same weight, and similarly you talk about longer skis as though I might have thought they were a disadvantage, whereas they are a very clear advantage.
I remember at one time they changed the rules about ski length and it made some difference to standings (I noticed that this was when the Japanese stopped being so good, but it may have been a coincidence.
I don't know how ski length was calculated before or is now. This is why I asked if anyone knew if it was based on height or weight or both. I would guess perhaps weight mostly and height to a lesser extent.
This is already a sort of handicap situation, so the new rule is just a new twist.

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PG, There's a 'simple' formula here (section 15.2)

Of course, it's assuming columbic friction between the skis and snow and neglecting any viscous forces resulting from the friction generated film of meltwater the skis glide over Shocked

. Anyone else get an 'o' level in physics?

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There wasn't an edge to my comment and I did not mean to sound cynical BTW. I did not even notice your mistake,laundryman.

What's puzzling to me is the phenomena(well, when I was at school, anyway) of static and dynamic friction, why rolling friction is greater than sliding friction etc, etc Has anyone claimed the Nobel Prize for Physics for that yet or was I not paying attention(as usual) at school?

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Kit Wong, I'm not sure that there is a well-developed theory of the different types of friction between solid surfaces, but then again I've not been tracking the field (it's difficult to maintain momentum wink

). This page, with the others around it, has got a pretty decent treatment.

I've got a way of thinking about static and dynamic friction. Friction is caused by bumps and pits on the surface engaging with other. Obviously these are usually distributed randomly in two dimensions but, to make things simple, imagine both surfaces have a regular set of ridges set some distance, d, apart. If the two surfaces are brought together with the ridges parallel, the ridges will be a maximum of d/2 apart (on average d/4). Suppose it's the former, that one of the surfaces is held in position, and that a force F is applied along the moving surface, at right angles to the ridges. If the moving object has mass m, it will accelerate at a rate of F/m, according to Newton's second law. Its velocity when the ridges bump into each other is given by v²=2as (one of the equations of linear motion) where s is the distance travelled. In this case s=d/2, so v²=2.(F/m).(d/2). Simplifying, v²=Fd/m. Now, the kinetic energy of an object is given by E=mv²/2, so in this case the kinetic energy is E = (m/2).(Fd/m) or E=Fd/2.

Assume this is just enough energy for the two sets of ridges to ride over each other, so that by the time they have, the moving surface has lost all its forward motion. With the force F still applied, there is now a whole d over which it can be accelerated before the next collision, so that it acquires twice the kinetic energy as previously (Fd), comfortably enough for the ridges to glide over each other. It's as if the resisting force (friction) has halved; now dynamic friction, rather than static friction. Obviously most real situations are more complex, but I think this model is a reasonable way of understanding why there is a difference.

I've not thought before about the difference between rolling and sliding friction, but I think it's probably to do with an apparent weight increase in the latter case. The frictional force is always proportional to the reaction perpendicular to the surfaces; that is, how hard they're pressed together, which causes the bumps and pits to mesh more tightly. If the surfaces are vertical, this is just the weight of the one object on the other, at least if they're sliding. If the top object is rolling though (imagine it's a bike tyre), then the weight will be augmented by the pressure of the front part of the tyre moving down into contact with the ground: put another way, it hurts more to have your foot ridden over than having a bike tyre (with bike and rider) placed upon your foot for the same amount of time (not tried it though).

Another dreadful pun: that's my theory, and I'm sticking to it (for the time being) wink

Before the good old days of ABS on cars, the only way of improving the stopping distance of a car with locked wheels under braking was to release the brakes, allowing the wheels to roll, before reapplying them. Is the reaction at the car wheel(s) the same for both cases and/or is it the coefficient of friction that's different? Sorry if this is going off at a tangent - this will be my last question!

You know it makes sense.

The reason a car takes longer to stop if the wheels are locked is because the rubber in contact with the road is scrapped off the tyre forming tiny pellets of rubber due to the heat build up during the skid which cause greater loss of grip the only way to stop this is to rotate the wheel again. On a wet road it is because the tyre aqua- plains on a thin film of water because when the wheel locks the tread of the tyre cannot clear the water away from the road surface by allowing the wheel to rotate the tread can again start working. Aqua plaining with wheel rotating can occur when there are large amounts of water for the tread to clear and the forward sped is to high to allow the tyre tread to clear the water. With regards to friction in both these cases the pellets of rubber and the water act as a lubricant and reduce to the friction during the slide of the tyre.

Otherwise you'll just go on seeing the one name:

Never seen anybody skiing down a mountain on a car tyre yet. The nearest I have seen is sitting inside a big fat inner tube (toilet position). rolling eyes

Must be pretty desparate if they do Aqua planning too? Laughing

A snowHead

A snowHead

as sparks fly wheel the wheels lock it could be an iron dust or iron oxide dust or both just an idea mind absolutely no proof wink

Well, the person's real but it's just a made up name, see?

I hesitate to put my oar in here, as I know almost nothing at all about physics. However as an observer, I should say that it would be virtually impossible to come up with any viable equation as there are too many variables in ski-ing.

PG, if you were thinking how handy it would be to be able to calculate how much better Hannah might do if 20Kg heavier, or what advantage the heavier girls have, it's entirely understandable, but the variables could change in between skiers, and thus negate any equation, since you probably couldn't either spot the variables that had changed or re-think the equation in time.

I hope this isn't too negative, but certailnly this would be the criticism of all the biomechanics papers I've read pertaining to ski-ing.

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easiski, it was more a reaction to something a parent said about the alleged lack of talent of his son. A beautiful skier to watch, but invariably off the pace in racing, although everything looked right. I mentioned the paper I had read, and suggested that on an untechnical, flattish course, his 40% handicap in weight probably equated to something like two seconds at least over a 60 second race.

I'm not particularly interested in arriving at a set formula to calculate the precise handicap, but would like to put something down in writing so that parents of children involved in racing can at least understand something of the processes involved.

I agree that you cannot even begin to factor in the variables other than on the likes of a straight flying K course, but my experience of watching racing certainly bears out some of the theories discussed.

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Skydivers use weight vests to equalise different fall rates between members of a formation team (fat streamlined blokes fall quicker than skinny lightweight birds etc).
It might be hard to get a reasonable amount of weight onboard a skier but it should be possible to do something along these lines, even if it's just a couple of Camelbaks.

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Kit Wong, just read your comment, I recall a programme years ago where they showed that trains do in fact have steel tyres fixed to their wheels with a heat fitting technique because under hard breaking the steel tyres get flat spots Shocked

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PG, OK. I had coffee today with a world top biomechanist and ran the idea past him. He told me that they're currently doing a lot of research into kicking footballs, flight, lift, bend etc. but that they have to do all the experiments in the lab so they can control the variables (including wind - speed direction & temp etc).

However, he thinks that one of the really clever physics guys should be able to come up with a sort of linked variable equation which you could use as rule of thumb. (At least that's what I understood). He said in the kicking research they have ended up with 3 equations, none solvable on their own, but taken together can produce a reasonable + or - result. He did say that (although he doesn't ski), it's clear there would be more variables in ski-ing than in football flight.

Hopefully some of the others on this forum will understand what I mean and be able to work it out.
Puzzled

There's a very nice discussion of energy dissipation mechanisms in skiing in, "The Physics of Skiing" by Lind and Sanders: http://www.amazon.com/exec/obidos/tg/detail/-/0387007229/qid=1101532152/sr=8-2/ref=sr_8_xs_ap_i2_xgl14/002-7688461-2969617?v=glance&s=books&n=507846&tag=amz07b-21 ( or the older edition ).

Anyone trying to discuss the effects of the various types of friction in skiing really needs to be completely familiar with their treatment of this subject.

While it may be possible to generate some reasonable, empirically based curve fits to the functions of interest (eg, terminal velocity as a function of skier mass), I'm not sure they would be all that useful because it's likely that small changes in unknown / un-measurable / rapidly changing variables (eg, the snow temp, the fore-aft pressure distribution under the ski, etc.) would greatly influence the results. This is the reason that I, for example, am not exactly jumping at the chance to develop such formulae.

Tom / PM

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Anathema to the scientist perhaps, but I was really only looking for a "rule of thumb" guide for practical purposes, ie to explain the processes at work to dispairing lightweight racers. This thread (and the one I started on Epic) have been very useful - thanks.

Help!!

Scientific help needed!

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