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# Scientific help needed!

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/mathmaticians 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.

This may well have come up here already, if so, apologies. I've searched, but to no avail...

Thanks!

### But I did sleep in a Holiday Inn Express last night

PG,

My degree had the word "Engineering" in it, does that count?

What are you looking for and why?

The simple version of the equation is F=MA. More mass = more force, more force = better ability to overcome drag (air)+friction(snow). The simulation shows that drag/friction does not increase proportionally to mass. Grossly oversimplified, this makes sense because the mass of the air and the snow does not change, therefore the drag and friction does not increase.

It seems simple if you think about a brick and a brick sized feather. It seems obvious that the brick will fall faster through the air. In a pool, the brick will fall faster to the bottom. And that brick will slide faster down a ski slope (if itwill slide at all). Substitute a hollow brick for the feather and the results are not as obvious, but they are the same.

Ah if life/skiing were only so simple. The study appears to make a few critical assumptions: that the ski lengths are the same, that the physical space occupied by the mass does not change and that the quality of the mass is the same. Most of us know that (in general) longer skis help a person go faster, but that they are harder to turn. A larger person will have more air drag for the same body position. Muscle is denser than fat. For racing, you're going to find that adding fat to get more mass to go faster either is not going to work or will at least suffer from diminishing returns. The bottom line is that there are a lot of other factors in the real world. This makes reality much more complex than the formulas presented in the study. So what use is a simpler formula going to be?
My interest is mainly with regard to children’s racing.

I do appreciate that other factors can, depending on conditions, influence or negate the effect of the drag/friction/weight element. Sure - strength, technique, lift, type of wax, length of ski used etc, all come into play. That said, from following the same groups of racers around race circuits over a period of years, it’s pretty clear that the lighter racers had a considerably better chance of making the latter pay in steep, technical, icy conditions. And again, you would have to assume that the heavier racer with identical strength/skill/balance would still come out on top in the latter conditions. Yes, longer skis, if the lighter racer can handle them, can help recover some of the difference, but you'd assume that a heavier/stronger racer with equal technique would be able to handle longer skis still.

My practical experience is that over an easy, softish, and in particular a flattish course, the weight issue can and does have considerable influence, assuming racers of equal ability, with a significant weight differential.

So I just want to factor in the weight ratio effect, all other elements being assumed equal. How is it useful? From the point of view of children's racing where I am involved, some parents of smaller kids – not to mention the kids themselves - can be disillusioned by the apparent gulf in speed. 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.

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!

### Well, the Holiday Inn just kicked me out...

PG,

The study says a 0.4% difference for what amounts to 6.67% difference in weight. Compared to all of the other possible factors, I'd say that's noise and not to worry about it - most of the time. As you've observed for icy conditions, the extra speed works to the detriment of heavier racers who are not highly skilled. But for the big difference in weight example, if Physicsman will check my math, the 30KG racer going against the 55kg tubbo would have a 3-5% disadvantage. For the flat easy course at racing above the Nastar level, that's a tough hurdle to get over.

You could use the results of this study as gospel and create a formula like:
(((W2-W1)/W1)/.067) * 0.4 = handicap percentage
W2=heavier racers weight
W1=lighter racers weight

But I'd caution that there are way too many assumptions (e.g. my engineering skills are a little [cough] rusty) made to assume that this calculation was fair.
Just set steep icy courses
The Rusty,

You seem to be on the right track; however, with the formula you suggest you'd end up with a nightmare for anyone trying to calculate it.

You might want to try a handicapping chart. A chart that gives a percentage based upon what the individual weights only compared to a fixed weight. Not a comparison to any other skier. Handicaps would be positive and negative. The lighter kids get a boost the heavyweights get a true handicap.

If you pick the comparison weight as 60kg then a typical chart would be as follows (using your formula):

Weight (kg) (Handicap (%))

30 (5.97)
35 (4.26)
40 (2.99)
45 (1.99)
50 (1.19)
55 (0.54)
60 (0)
65 (-0.45)
70 (-0.85)
75 (-1.19)
80 (-1.49)
85 (-1.76)
90 (-1.99)

All assuming that the original studies and the equations are correct and fair.

You might have to fuddle the positive and negative signs. (Lighter kids would have time subtracted while heavier would have it added.)

That's this engineers take on it.

Remember, Engineers are faultless and acurate!
T2,

Cool! The Holiday Inn wouldn't let me stay another night. Yeah, .... that's why I stopped with the "simple" formula. Yeah - that's the ticket.
Quote:
 Originally Posted by PG 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/mathmaticians 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. This may well have come up here already, if so, apologies. I've searched, but to no avail... Thanks!
I haven't got the formula but I can add a little on the 1st principles effects at play. Its all about cubes versus squares.

As an 3-d object (e.g. a person) get generally bigger its weight increases in proportion to the cube of its height but its air resistance increases according the surface area its presents, which is proportional to the square of its size. If you chip a tiny piece off a brick it will fall to the floor slower than the brick itself. Similarly, all other things being equal, a large skier's increased mass will more than componensate for his increased wind drag.

Interesting a reverse effect applies to strength. Weight increases according to a cube law whereas strength (which is approximately proportional to the cross sectional area of a muscle) increases with a square law. This is part of the reason why gymnasts are usually relatively small. You can see this affect in the gym. The really big guys are all benching huge weights but they are normally shy of the pull up bar (especially the wide grip).

Taken together this will mean a heavy racer will benefit more from grativity versus wind resistance but will find it harder to supply the necessary leg strength and endurance for good technique.
The Rusty,

That's what is great about the forums. We can spring off each other's ideas and come up with some cool stuff. (None of us is as smart as all of us.)
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