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The Spokesman-Review Newspaper
Spokane, Washington  Est. May 19, 1883

Bill Jennings: The physics and mathematics of skiing

By Bill Jennings Correspondent

Math is really hard work for me, but I’ve always enjoyed physics. Some physicists who write have a knack for making the complex simple to help you understand their universe, such as how Einstein’s famous equation E=MC2 explains why if a baseball is thrown at you hard enough, it will knock you down. That’s why I couldn’t resist ordering The Physics of Skiing: Skiing at the Triple Point, by David Lind and Scott P. Sanders.

According to the authors, they chose the title of their book for two reasons. The first being skiing works best at the triple point of water, a magic zone where water’s three possible states – solid, liquid and vapor – coexist.

Their second reason is to share with readers their true joy, made possible by an understanding of the physics involved in the sport: a triple point where increased knowledge of the “how” and “why” is joined with the “wow” of skiing.

The authors’ dissection of skiing minutiae ranged from the polymerization of water molecules into snowflake crystals to the kinetics and kinematics involved with the path of quickest descent – a challenge known to physicists as “The Brachistochrone Problem.”

The content is enough to leave most of us glassy eyed. Nevertheless, I absorbed some interesting revelations and “fun” facts.

For example, I’ve always wondered how the turning radius of a ski is calculated. It’s an approximation of the radius of a circle passing through the tip, tail and waist of the ski edge. The equation looks like this: R = C2/SC; where R (the turning radius) equals C (the length of the ski in contact with the snow, squared) divided by SC (the side cut).

Another equation gives you the number to plug into SC: 1/4 (S – 2W + T): S (the shovel width) minus two times W (the waist width) plus T (the tail width), times one quarter.

Fortunately, someone does the math for you. Look on the ski and there it is. My ski’s turning radius is 17 meters. However, could awareness of a little bit of the “how” and “why” of a spec blithely tossed about by ski marketers enhance my “wow?” Maybe.

Ski instructors can make a simple act maddeningly complex. Wait until you analyze ski technique through the lens of Newtonian Mechanics.

A quick review of Newton’s three laws of motion: 1) a body in motion stays in motion until acted upon by a force; 2) if a force acts upon a body in motion, its motion must change; and 3) for every force acting on this body, an equal and opposite force is acting against it.

When you ski, a lot of forces are at work on your body in motion (law No. 1). Those forces (law No. 2) include velocity, acceleration, mass, angular motion, torque, momentum, gravity – both normal and parallel to the slope – aerodynamic drag and lift, friction and last but not least, the reaction force of the snow (law No. 3).

Lind and Sanders provide a lot of diagrams and equations about how these various forces act upon a skier. My head spun.

If I could ask Sir Isaac for help, he might say, “Your mass will accelerate down the fall line. To control your acceleration, you are going to initiate torque with your ankles, knees and hips to tip the ski on edge. This force will impact angular motion, velocity and momentum, causing your skis to turn. These and other myriad equal and opposite forces acting and reacting with themselves, gravity, terrain, your equipment and yourself will propel you down the mountain. Consider yourself enlightened.”

Thank you, brother Isaac. Fortified with knowledge about the how and the why of skiing, I headed for the hills.

As I skied, my thoughts went to all the advanced physics my brain calculated, harnessing the forces of nature and making adjustments hundreds of times per second as it solved The Brachistochrone Problem with such incredible precision on the fly. Wow.