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Reprints From The Professional Skier

Fall 1996 - "Longer Line = Shorter Time " by George Twardokens

This article is reprinted from The Professional Skier. All copyrights apply. Please see our copyright and disclaimer notice page.

Conventional wisdom holds that the shortest distance between two points is a straight line. While this is true, it may surprise you to discover that the fastest route between two points is not necessarily that hallowed straight line.

If, as a ski coach, you think your racers can turn in faster times by taking a straight line between gates, you may be interested in a study I conducted that showed that the fastest line between two points on a ski slope is very often a cycloid curve. A cycloid curve is the path followed by a point on a wheel that rolls along a straight line (fig. 1 below).

This study, conducted at California's Squaw Valley Ski Resort, put a skiing spin on a centuries-old theory that is known in variational calculus as brachistochrone (from the Greek word for "shortest time:') This theory, first observed by the Swiss scientist Johan Bernoulli in 1696, holds that the path of a particle sliding without friction from A to B under the force of gravity is not a straight line, but a cycloid curve. In fact, the calculus of variations had its beginnings in the solutions to this very brachistochrone problem.

In Short

Wishing to find out if brachistochrone had bearing upon the world of ski racing, I designed a simple on-slope experiment conducted by a research team that included myself, Rolf Paulsson, of the Institute of Theoretical Physics in Uppsala, Sweden, and several skiers experienced in coaching and racing.(1) Simply put, we set out to compare the time needed to traverse on skis in a straight line between points A and B to the time needed to carve a ski turn that follows a cycloid curve connecting the A and B points (fig. 2 above.)

Slope And Subjects

The experiment was performed on a 30 to 40-degree salted slope at Squaw Valley Ski Resort in April 1988 with skilled skiers and sophisticated timing equipment. The straight traverses and cycloid curves for three different distances were marked in the snow with colored dye. Paulsson had calculated and graphed the curves used in the study before the experiment (fig. 3 above). Those wishing to delve deeper into the math as it applies to this study will find it in the appendix.

Using slalom skis, three experienced racers skied the straight lines and the curves in separate test runs. Skier Steve Gould weighed 154 pounds and skied a 200-centimeter ski, 194-pound Sean Walkerly skied a 205-cm ski, and 114-pound Alenka Vrecek skied a 195-cm ski. David Leer, another member of the research team, timed them. The racers used a gravity-start procedure in which they started from a standstill slightly behind the wand.

The Results

When the slope was prepared for the experiment, numerous bystanders said, "It will never work!" Fortunately, it did. As you can see from the results listed in Table 1 below, we found that in 15 out of 18 test runs, descents over the longer path outlined by the cycloid curve were faster than the descents over the straight line traverses. In each of the runs that produced a shorter time on the traverses rather than on the curve, we observed considerable skidding of the skis instead of the desired carving. The study suggests that, indeed, the shortest line between the gates may not be the fastest.

Other Studies

To the best of my knowledge, this study represents the first controlled experiment addressing the brachistochrone concept in real skiing situations. Subsequent studies seem to support these findings. In a 1991 French study based on experiments and mathematical calculations, Gilbert Reinisch reported on the optimal trajectory for a giant slalom turn. According to his findings, the optimal trajectory between point A and point B is part of a cycloid curve that appears to be relatively flat according to initial velocity of the racer at point A.(2)

Computer calculations performed by Stewart Ambler in 1991 indicate that the optimal path of a cycloid curve flattens as initial speed at point A increases.(3)

What About Wind And Friction?

As you might expect, wind that pushes the skier can be a strong propelling force. Similarly, air resistance, which increases with the square of the skied velocity, can be a factor. In this study, neither wind velocity nor its direction were evaluated. However, one can hypothesize that, on the average, wind influenced the straight runs about the same as it influenced the cycloid-curve runs.

In addition, we neglected friction in calculations related to this study. Modern ski bases are so slick (coefficient of friction as little as .02) that they reduce friction enough to bear out Bernoulli's theory on brachistochrone. A consideration, of course, is how closely the racer can achieve the conditions required for the study's conclusion to be valid. In principle, the pure carved turn has the least friction and, thus, will best approach the cycloid.

Discussion

Casual analysis may suggest that the advantage of the curvilinear path - which at the outset of the descent is closer to the fall line than the straight traverse line-is that it affords more acceleration and, therefore, more velocity near the start. However, this observation must be weighed against the advantage of the shorter distance covered by the straight traverse's path. At this point, intuition can no longer be helpful and there remains only the mathematical method to shed light on the debate.

The mathematical solution to the concept of brachistochrone is well known in mathematical circles. It was published by Johan Bernoulli, Isaac Newton, and Gottfried Leibniz, although their techniques for arriving at the solution varied.

Today, virtually every text on classical dynamics or variational calculus deals with the brachistochrone problem in theory, but application of this theory to sports is rare.

Mathematical theories aside, this ski-specific consideration of brachistochrone contains "news you can use." During a 1990 National Ski Coaches School at Mammoth Mountain, California, many assembled coaches doubted that cycloids could succeed. An impromptu reenactment of the experiment was arranged. The coaches were amazed when the cycloid was faster than the straight-line traverse in most of the trials.

Conclusion

In this short study (a labor of love conducted with no grants), many essential controls and measurements, such as those relating to wind speed and direction and the steepness of the slope, were omitted due to financial constraints. And without a doubt, many pertinent facts remain to be discovered. But if, in the future, the whole truth emerges, it may very well throw a curve at those who believe the fastest path between two points is a straight line.

At the very least, the empirical results obtained in this research suggest some practical advice for competitive skiing: the shortest line of descent between the gates may not be the fastest. On the contrary, the rounder line, in which the racer often completes the turn well below the gate - almost late and off line - may, in fact, be the fastest way to ski a course, particularly in GS, Super-G, and Downhill races.

References

1. George Twardokens, "Brachistochrone (i.e., shortest time) in Skiing Descents," in Proceedings of the Eighth International Symposium of the Society of Biomechanics in Sports (Prague, Czechoslovakia, 1990), 205-209.

2. Gilbert Reinisch, "A Physical Theory of Alpine Ski Racing," Spectrum der Sportwissenschaften (Austrian Sports Science Association) 1, (1991): 26-50.

3. Stewart Ambler, "Cycloid Graphs and Newer Method of Computer-Generated Optimal Path" (P.O. Box 1705, Longmont, Co. 80502,1991).

George Twardokens is a professor of kinesiology at the University of Nevada-Reno. He is the author of Universal Ski Techniques as well as numerous articles on the biomechanics of skiing. A 25-year member of PSIA and former PSIA-W examiner, "Dr. T" is a Certified Level III alpine instructor at California's Alpine Meadows Ski Area.