Introduction

Endurance sports such as running and cross-country skiing share many characteristics which make them attractive for alternative season activities. The joy of running forest trails is matched by skiing through the woods on tracks that roll up and down adapting to terrain. Running such trails can be physiologically demanding but involves relatively simple technique. The subtle adjustments of stride length and frequency, of foot placement and arm motion are rather intuitive and come naturally to most runners. In contrast, skiing such wonderful changing terrain requires a variety of complex techniques smoothly linked together. Both running and cross-country skiing demand large metabolic capacities; race results are certainly determined in large part by competitor physiology. But in skiing, technique and equipment probably play a much greater role in affecting performance. This chapter will explore the mechanical side of skiing--looking to explain the subtleties of technique, characteristics which drive performance, interactions of equipment, technique, and economy of motion. To understand the mechanical side of skiing, it will be helpful to use the tools that engineers and physicists use for analysis of inanimate systems.

Biomechanics applies engineering methods to analysis of human motion; we'll explore what biomechanics can explain about skiing technique and performance. Mechanics can be broadly divided into two approaches: "kinematics" deals with descriptions of motion using characteristics such as displacement, speed, and acceleration while "kinetics" deals with the causes of motion such as force, torque, and energy. To understand human motion it is often helpful to start with kinematics so that one can quantitatively "picture" movement patterns. But ultimately, kinetic characteristics are what drive motion and which must be determined to approach understanding of complex movement patterns like those of cross-country skiing.

Mechanical Principles

While the British are not often thought of as a skiing "powerhouse," performance in skiing is best understood through the observations and insights of the famous non-skiing British scientist Isaac Newton. His contributions to classical physics are well known to any general physics student, but restatement and discussion of these physical ideas in the context of cross-country skiing can clarify the fundamental characteristics which affect skiing performance. Newton's laws of motion are often memorized as dealing with inertia, acceleration, and action-reaction. Important concepts behind these phrases will require some explanation. We'll keep the skiing context in mind but recognize that the principles generalize to any motion.

Newton's Law of Inertia. A skier (or any other object) in motion will continue moving in the same direction at the same speed unless some external force (like gravity or friction) acts to change the motion characteristics. Visualize a skier gliding downhill and then across flat terrain on icy, fast tracks. A tailwind blowing behind the skier minimizes air resistance. The skier feels able to glide forever without slowing down. That's inertia. If friction across the snow were really zero and the wind blowing enough to eliminate air resistance, the skier's motion would in fact continue at the same speed and in the same direction forever! Skiers don't naturally slow down when gliding. It requires external forces like friction to accomplish that. Newton's second law which follows describes how much an external force affects a skier's motion.

Newton's Law of Acceleration: F = ma. A force (F) acting on a skier will cause an acceleration (a) of the skier in the direction of the force and proportional to the strength of the force. The acceleration will be inversely proportional to the skier's mass (m). This relationship which is so simply summarized in the equation F = ma is perhaps our most important tool for understanding skiing mechanics. Re-visualize the skier gliding on flat tracks. Friction on the skis is a force acting opposite to the direction of motion. It will cause a decrease of speed which is often called a negative acceleration or a deceleration. The severity of the deceleration depends on the magnitude of the frictional force and also on the skier's mass.

Now put a strong tail wind back into your picture of the gliding skier. If the wind speed is greater than the skier's speed, it will exert a force pushing the skier from behind. At some speed this "push" from the tail wind could exactly match the frictional force of skis gliding on snow. What would happen? The two forces are in opposite directions and of the same magnitude or strength. Where the snow frictional forces would decelerate, the tail wind would accelerate; the forces would balance, they would combine to be of no effect. This illustrates a deeper meaning of F=ma. Acceleration of a skier results from all the external forces acting; some may cause negative, some positive acceleration. The total force with contributions from all external forces is what matters. In our example, the snow frictional force and the pushing force of the wind are of equal strength and in opposite directions. They combine to zero force. From F = ma, if total external force is zero then acceleration must be zero; the skier will continue gliding along at constant speed. Notice that this is just a different way of getting to the idea of Newton's first law.

Total Force is a slightly more complicated idea when the forces acting on a skier are not collinear, that is not acting in the same direction. Returning to our gliding skier, if the wind were blowing past the skier just as before but from the side instead of from behind, the snow friction and wind force would not combine to be zero. Some more complex combination determining the total force would be acting to change the skier's motion. This requires knowing the directions of all the forces involved and combining them using the mathematical technique of vector addition. This will be described in more detail after introducing Newton's third law.

Newton's Law of Action-Reaction. A skier's push against the snow is matched with a reaction force of equal magnitude but opposite direction of the snow pushing on the skier. Muscle activity and body motion can create forces against the snow ("action" in Newton's terminology). Such actions are always paired with reaction forces applied to the skier. Note that these two forces do not "cancel" each other out as they act on different objects. The action force is applied to the snow and earth (very large mass and small acceleration) while the reaction force is applied to the skier (relatively small mass and substantial acceleration). It is reaction forces which largely determine a skier's performance.

Like other forces, reaction forces are vector quantities. The skier of figure 2 pushes down through the poles and applies a force against the snow. The snow reaction force is of the same magnitude but aimed up the pole and ultimately to the body through the hand and arm. This reaction force aimed along the pole is composed of two parts, a horizontal component and a vertical component. Together these components make up the "resultant" reaction force; separately they affect motion in the horizontal and the vertical directions. The greater the horizontal component, the greater will by the skier's acceleration in the forward direction. This is called propulsive force. The vertical reaction force component affects motion up and down only and does nothing to propel the skier forward.

The relative proportions of the horizontal and vertical reaction forces depends on the angle of the resultant reaction force with respect to horizontal or vertical. In figure 2 this depends on the pole angle and how inclined it is from vertical. As the pole inclines away from vertical, the horizontal component, the propulsive force, becomes larger while the vertical component becomes smaller. This relationship is illustrated in figure 3 at the beginning and near the end of a poling action.

Skier generated reaction forces at the poles and skis are a competitor's primary means of moving down the ski tracks. However, several other forces also affect the motion. Gravity is a large force which is determined by a skier's mass and which is always directed vertically downward. On downhills, a proportion of the gravity force (or weight) of the skier is aimed in the forward direction and acts to propel a skier down the slope. Figure 4 illustrates how the steepness of a downhill affects the propulsive component of gravity. In a similar manner on uphills, gravity acts against a skier's forward motion. But in either case there is little that a skier can do to change the gravitational force acting to slow or to propel him down the tracks. In contrast, frictional forces of skis on snow and the body passing through air can be affected by skier technique and equipment.

Air resistance or air drag is a complicated force which depends on a skier's shape and size, on the atmospheric pressure, and on the relative velocity of air passing the skier. Origins of air drag force are ultimately due to pressure differences on the front and back of a skier's body, but we need not go into those details here. The tuck position seen in figure 4 is an effective downhill technique because it streamlines airflow and it minimizes frontal area of the skier. We'll explore these relationships in more detail later in this chapter. Relative velocity of air passing the skier can be positive or negative in direction as can the air drag force itself. A headwind will cause air drag forces resisting a skier's forward motion while a tailwind blowing faster than a skier's forward motion will help propel that motion. Hence air drag, like gravity, can be propulsive or resistive in direction.

In contrast, ski drag force is always acting against a ski's forward motion. Numerous factors such as snow conditions, wax, and ski stiffness affect the magnitude of ski drag force and these will be explored in more detail later in this chapter and in other chapters of this book.

This collection of forces we have been describing are illustrated in figure 5. Newton's second law tells us that it is the total force resulting from adding all the external forces together which determines a skier's motion. Let's estimate each of the forces shown in figure 5 and see how they combine to determine the skier's acceleration at the moment in time illustrated. The skier shown had a body mass of 60 kg and therefore weighed about 600 N which is the gravitational force acting vertically downward. Skiing at 5 m/s she had a horizontal drag force resisting forward motion of about 10 N. Ski drag force depends on the snow/ski coefficient of friction and on the force pressing vertically down on the ski. So let's first determine the reaction force at the ski. Using instrumented skis, reaction forces at the moment illustrated during glide are about 80% of a skier's weight or about 500 N in this case. These various forces are illustrated in figure 6 where it can be seen that the reaction force is angled slightly backward (about 5° or so from vertical), hence there are horizontal and vertical components of the reaction force. These are proportional to the sine and cosine of the 5° angle with magnitudes shown on the diagram. From the vertical reaction force we can estimate the ski drag force to be about 50 N. Finally, the pole reaction at this point in a stride is about 25% of a skier's weight or about 150 N in this case. With the pole angled forward at about 25° from vertical, horizontal and vertical components of the poling force can also be calculated using the sine and cosine trigonometric functions. The force components shown in figure 6 can be grouped into those directed horizontally and those directed vertically.

In the horizontal direction if we consider the skier's forward direction as positive, the horizontal pole reaction force is +63 N while all the other forces are negative: air drag force (-10 N); snow drag force (-50 N); ski reaction force (-44 N). Newton's second law tells us that the sum of these horizontal forces determines the skier's horizontal acceleration at this moment of time. Thus F_horizontal = +63 - 10 - 50 - 44 = - 41 N. Further, knowing that F = ma, we can determine the skier's acceleration at this instant in time (a = F/m = -41 N / 60 kg ( -0.7 m/s/s). The skier was slowing down by about 0.7 m/s per second while gliding on that ski.

In the vertical direction if we consider up to be positive, the ski and pole reactions forces are positive (+498 and +136 N, respectively) while the gravitational force (skier weight) is downward (-600 N). Again from Newton's second law, the total vertical force will determine the skier's vertical acceleration. Thus F_vertical = +498 + 136 - 600 N = +34 N and from which vertical acceleration is a = F/m = 34 N / 50 kg ( +0.7 m/s/s. The skier had a slight vertical acceleration. Note that this vertical acceleration is independent of the air and snow drag forces which are horizontally directed.

This rather lengthy computational example has applied Newton's laws of motion at a single moment in time to determine the effect of external forces on a skier's motion. But we are interested in much more than such single moments in a stride. To determine the full dynamic stride mechanics would require repeating that process moment by moment throughout the stride based on measurements of the external forces. While such measurements are not easily obtained, instrumentation has been developed to record the reaction forces of skis and poles during skiing. Figure 7 illustrates the reaction force patterns for the diagonal stride technique of our previous example. Qualitatively assessing the magnitudes of these reaction forces, relatively low magnitude propulsive forces (y direction in the figure) are observed except for the brief "kick" of the ski and during later poling. These brief periods are the only times when skier generated propulsive force is of greater magnitude than the air and snow drag forces resisting forward motion and the only times during a stride when positive forward acceleration is generated.

The motion during diagonal stride used for the previous example and figures 5-7 is a relatively planar movement pattern which involves little side to side motion. Looking at the arm and leg motions and the forces in the sagittal plane (side view) is a good approximation to the full three-dimensional motion. Thus the pole and ski reaction forces shown in figure 7 did not include a component in a side-to-side direction. In contrast, the skating techniques involve considerable side-to-side motion as well as in the other dimensions. To measure the three-dimensional force components during skating requires knowing the direction of the reaction forces in space--a much more difficult determination than for the two-dimensional forces of diagonal stride. Figure 8 illustrates this for the pole reaction force resolved into propulsive, vertical, and side-to-side force components and would require knowing the force applied along the axis of the pole and the pole's three-dimensional orientation in space.

The pole orientation illustrated in figure 8 is commonly observed during some skating techniques and results in a substantial force component in a side-to-side direction as well as vertical and forward directions. As orientation of any of the reaction forces change, the effectiveness in propelling a skier forward will change in proportion to the propulsive component of force. Thus, each pole and ski orientation can influence the effectiveness of reaction forces generated by the skier.

Newton's laws tell us that the total force in each of the three-dimensional directions will determine the skier's acceleration in that direction. Performance is determined by forward motion, hence those components of reaction force in the forward direction are crucial for a skier to generate and the negative components due to drag are equally crucial to minimize. However, motion in the other directions is also necessary for many of the ski techniques and cannot be eliminated. The following sections of this chapter will systematically discuss the classical and skating techniques with a particular focus on how each technique's kinematics interact with the forces in each direction.

Next Section

Figure Captions for this section:

Figure 1. A skier in motion will continue at the same speed and in the same direction unless there are external forces acting. This is the idea of inertia. Gliding across flat terrain if air resistance and friction acting on the skis were zero, a skier would continue moving forever.

Figure 2. Poling force is a vector quantity. The resultant force is composed of a vertical and a horizontal (propulsive) component. The magnitude of the components depends on the angle of the pole with respect to vertical and horizontal. With a pole angle of 20 degrees from vertical and a 200 N force along the axis of the pole, about 68 N of propulsive force [200 x sin(20°)] is generated.

Figure 3. Vertical and propulsive force magnitudes depend on the applied force along the pole and on the pole angle. With the same applied force, as the poles are inclined away from vertical, the proportion of propulsive force increases and poling becomes more effective in propelling the skier forward.

Figure 4. Gravitational force acts to propel a skier downhill. The magnitude of the force depends on the slope. A skier weighing 700 N (mass of about 70 kg) on a 10 degree downhill has a gravitational force propelling the motion of about 120 N [700 x sin(10°)].

Figure 5. Motion of a skier depends on all the forces acting at any moment in time.

Figure 6. Finding the total effect of all the forces acting on a skier at some moment in time depends on summing all the forces in a vectorial manner. Each component of force must be known and appropriately summed. The plus and minus signs are indicative of the force directions and are conventionally taken to be upward positive, downward negative and to the right positive, to the left negative. Vertical force components are combined and horizontal forces are combined.

Figure 7. Pole and ski reaction forces during diagonal stride on uphill terrain. Such forces are often expressed in multiples of body weight (BW). So for example, at pole plant vertical poling force is about 20% of body weight. During kick phase, vertical reaction force at the ski is about 2 times body weight. Figure adapted from Komi (1987).

Figure 8. Three dimensional force components are required to analyze skating kinetics. The components depend on the orientation of the pole in space. This requires two angles to describe and considerably complicates force data collection compared to two dimension analysis.

Next Section