Originally Posted by Martin Bell
Out of interest, PM, are these velocity-generated forces of interaction with the snow dependent on the skier's kinetic energy or his/her momentum? My vague recollections of high school physics (George Watsons, daslider) tell me there are different formulae for those.
Very, very good question. The dynamical aspects of the analysis of the dynamics of a group of objects interacting with others (eg, a ski with the skier and snow) are essentially axiomatic. In other words, they are usually little more than separate F=m*a equations for each object, with the equations written in some appropriate coordinate system (eg, linear, spherical, cylindrical, etc.), and in some convenient frame of reference (eg, stationary with respect to the earth, in uniform linear motion, in steady circular motion, located on one of the objects, moving with the center of mass of all the objects, etc.). The simplicity of these equations (and underlying concepts) is why I can be so completely dogmatic and unambiguous about these aspects of just about any dynamics analysis.
OTOH, writing F=m*a a bunch of times does not provide a complete, solvable set of equations with which you can turn the mathematical crank and find out how the objects will move. The missing part of the puzzle is a mathematical description of the interactions between the objects under analysis. These formulae are usually termed the “materials equations”, “force fields”, or “constitutive relations”, and are almost always the most complicated part of a statics or dynamics analysis. Sometimes these relations can come from elaborate side analyses (eg, quantum mechanics), but often, they are little more than curve fits to empirical data. The latter is the case for a ski interacting with the snow.
While one typically can’t write down a complete formula for the materials equation for a given type of interaction, there are procedures (one of which is called “dimensional analysis” – Google this phrase plus “fluid mechanics”) which can at least suggest which sets of variables must appear in these equations, which variables can be immediately eliminated, the exponent to which the variable must appear, etc.
The question you asked about whether the ski-snow interactions are best described in terms of momentum or kinetic energy falls exactly into this category.
Consider a ski held in a fixed orientation (edge angle, angle of attack, and tip up-down angle) and dragged over or through the snow at some velocity (ie, both speed and direction) while being pressed into the snow by a specified weight. The sideways force of the ski on the snow (either drag or holding) is then measured.
Clearly, the direction the ski is traveling (everything else held constant) will make a big difference in the drag the ski will experience. Skidded sideways in one direction, a ski will tend to ride up and over the snow, while in the opposite direction, it will dig into the snow.
Since kinetic energy (KE = 1/2 times mass times speed squared) is a scalar, not a vector, this means that it contains no information on the direction of motion of the object (the ski), so we can immediately rule it out as a fundamental quantity in a description of ski-snow interactions. Of course, one could combine other more fundamental variables like mass and velocity to construct kinetic energy as a variable which could be used in such a materials interaction formula, but you would still always need to have mass and vector velocity around in any equation to fully specify the ski-snow interaction.
Momentum (mass times vector velocity) is a more promising variable since it does contain directional information. The real question about it is whether mass and velocity can always appear linked together (as they do in momentum), or need to appear separately in order to reasonably describe the interaction of the ski with the snow. A moment’s reflection should make it apparent that mass itself is not a fundamental parameter in determining the (skier+ski) interactions with the snow, but is the vertical component of the force with which the skis are being pressed into the snow that will determine the sideways drag force. The distinctions here that have to be thought about include sloped surfaces, and is it really mass or downward force (which might easily include centrifugal components) that is important. Clearly it is the latter.
So, after all this analysis, the short answer to your question, “…are these velocity-generated forces of interaction with the snow dependent on the skier's kinetic energy or his/her momentum?” is that these forces can, of course be said to be dependent on either variable, but the simplest, most useful, and most concise description will use neither of these combination-type variables but use (a) downward force, (b) velocity, and (c) ski angular orientation as the fundamental variables. If I remember correctly, these are exactly what Twardokens’ father used in his famous sled experiments.
Originally Posted by Martin Bell
…Look forward to seeing the forces table. Any World Cup ski tech can tell you that considerable friction takes place at the section of the ski base that is adjacent to the metal edge, causing the dreaded "edge burn".
My table will provide a very accurate number for the total normal (ie, perpendicular) forces on the base and side of the ski, but it won’t say anything about whether the upward force on the base is concentrated near the edge or spread out across the width of the base. My analysis is intentionally limited in scope in this way in order to make it extremely general, accurate, and use the absolute minimum number of parameters and assumptions. An analysis which would predict the distribution of forces across the width of the ski would require an accurate model of the mechanical strength of the snow underfoot. Unfortunately, that is empirical data which varies greatly from one patch of snow to the next, so I didn’t even try to include it.
Tom / PM