THE PHYSICS OF SLALOM WATERSKIING

By Benoit Lance

Did you ever wonder about the top speed, acceleration and rope tension you reach when going through the slalom course? Did you ever think about a scientific approach that would teach you about your performance?

As a recreational skier with a physics background, I began to thinking about this topic since the summer of 2019, after a slalom training at “Le Plan d’Eau”, in France. I was surprised that so little was written about waterskiing, as opposed to alpine skiing, for which several articles and books are written. Amongst the four studies I identified, one was dedicated to waterski jumping, in a chapter of a French book dedicated to the physics of sport. Another, the most recent one, was performed at the University of Guelph, in the province of Toronto. It considered the development of measuring systems impacted on the skier and his/her ski and was focused on slalom. I first concentrated on slalom modeling and decided to implement a model on an EXCEL worksheet.

After getting initial insights by simple rules of three, I thought about slalom like a moving harmonic oscillator, if you remember your physics lessons ☺ Although this is interesting to get some insight about the speed variation throughout the slalom course, this approach provides nothing about the forces involved.

And so, what about them? Those forces are driven by hydrodynamic effects, all being proportional to the square of the skier speed. One identifies several contributions, taken independent from each other for the purpose of simplicity: the water friction, uplift and wake drag contribution, that is the force applied by the skier while adopting a side leaning position. All forces are assumed to be applied on both skier and ski, resumed to a single point at the handle. This is a 2D problem, for which the forces are resolved into tangential and radial contributions: the tangential contribution is responsible of the acceleration and deceleration of the skier, while the radial one is responsible of the rope tension the skier has to sustain, especially during the acceleration phase. The figure below illustrates the forces and the chosen referential, during the cut phase. In order to mathematically express those forces, the ski orientation has to be expressed through 3 angles: the pitch, heading and roll angles:

The pitch angle is measured between the ski in a static position (no slalom traverse) and the water surface It can be expressed as a function of the skier speed;

The heading angle quantifies the way the ski is directed towards the coming buoy and is measured between the ski and the boat directions. It is computed in order to maximize the tangential contribution of the force applied by the skier

And the roll angle is the one made by the ski from the water surface when the skier adopts a side leaning position. The maximum value of this angle is chosen by the user, while the way the angle is varying is implemented in the application: the angle is kept to its maximum value during half of the traverse (i.e. acceleration phase up to the boat channel), then evolving linearly up to the opposite value at the following buoy.)

The equations of motion have been implemented into the EXCEL worksheet:

But let’s talk about the results!

The figure below illustrates the handle and the ski position during the slalom course, here simulated for a boat speed of 58 km/h, rope length = 16 m, a skier size of 1.8 m and weighing 80 kg. It has been calculated to take into account the body motion along the radial direction during the pre-turn phase, otherwise you can forget performance with a rope length below 12m!

The simulations were compared to the few results that were available: good agreement with the measurements (speed, acceleration, and rope tension) of the Canadian study, considering recreational skiers completing the slalom course with a boat speed of 52 km/h and a rope length of 18.25 m and also comparisons with rope tension measurements reported during Slalom Pro Tour events (special thanks to Vincent Stadlbaur!), from which it seems that the simulated rope tension is slightly overestimated (rope tension higher than 5x the body weight for a rope length lower than 10.75m, at the end of the acceleration phase).

When a skier adopts a too-high roll angle, the rope tension is unnecessarily high, but also the overall distance traveled by the skier: the skier tends to turn too far away from the buoy and excessive force and acceleration must be created by the skier to get back the lost distance from the previous buoy.

Of special interest is the speed dependence with the rope length: the figure below illustrates the average, minimum, and maximum skier speeds, along with the boat speed of 58 km/h, for various rope lengths from 18.25m and down to 10.25m. While the average skier speed is ~ 16% higher than the boat speed, whatever the rope length, the minimum and the maximum skier speeds are respectively decreasing and increasing with shorter rope length, up to extreme values of ~ 30 and 105 km/h. This suggests that the maximum tangential acceleration also increases with a shorter rope length, from ~ 2 up to 3 “g”, located just after the turn. Of course the tangential and the radial forces display the same increasing trend with a shorter rope length, the increasing rope tension being a price to pay for higher performance.

The study is still on-going and improvements are expected: measurement campaigns are envisaged in order to (i) fix some hydrodynamic parameters that were somewhat empirically chosen so far, and (ii) perform comparisons with the measurements. Further options have been added to the application, especially the simulation of the gate and the roll angle variation that the skier can choose. So if you are interested, we can speak about the gate in a next article. Unless you are more interested in reading the simulation tool I developed for jump which will give you useful insights to improve your jump performances!

Should you be interested in using the application, please let me know.

Anyway, have fun on your ski, Ben