The Physics of Cross-Country Skiing

The Physics of Cross-Country Skiing

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Cross-country skiing is as much of a competitive sport, as it is a back country one. Cross-country skiing is enjoyed by people of all ages, and can be relatively inexpensive. There is no need for lift tickets and with a little maintenance equipment can last for decades. As a result of its broad audience, many people don't realize that physics plays a large role in cross-country skiing. This web page was designed to briefly describe some of the concepts behind the physics of skiing, and give a basic understanding of both the sport and the science.

Friction is obviously an important factor of cross country skiing. On one hand, friction is necessary because without it a skier wouldn't be able to ski up hill or even move on flat surfaces. However, when racing, skiers prefer to have the smallest friction force working on them possible.

In order to reduce the frictional forces which would slow them down, skiers wax the bottom of their skis. The types of wax which they apply to their skis have different functions. The hot wax which is applied to the entire ski reduces the friction between the ski and the snow. This allows the ski to glide on the snow and gives the skier more distance for each stride.

Another type of wax is applied to the "kicker". The kicker is the area under the bindings. In other words the kicker is the area where most of the gravitational force of the skier is applied. Skiers apply a type of wax to the kicker that will cause the friction coefficient to increase. This allows skiers to be able to push off and ski uphill.

In order to begin their outdoor adventure, a skier must first face the forces of static friction. Static friction is the force that keeps the skier at rest. As the skier overcomes the static friction there is a point where the coefficient of friction is greater than that of the kinetic friction that resists the skiers motion. It is clear to see this concept in the figure below.

From the figure above, it is also easy to see that the kinetic friction remains almost constant for a range of speeds. This kinetic friction is the force which slows the skiers down after they start moving.

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Without kinetic and potential energy cross country skiing would be extremely boring. There would not be the frustration of trying to maneuver up hill, only to slide down backwards in a failed attempt. Skiers would also miss out on screaming down those hills and crashing into a snow bank.

Lets assume that energy is conserved when a cross country skier decides to conquer a mountain. When the skier is at the bottom of the hill their kinetic energy is at it's maximum and their potential energy is at zero. As the skier maneuvers up the hill their kinetic energy slowly decreases at the same rate that their potential energy increases. When the skier is halfway up the hill the kinetic and potential energies are equal. A few hours later when the skier has reached the top of the mountain, their potential energy is at its maximum and their kinetic energy is zero.

Kinetic energy is one half times the mass times the velocity squared. Kinetic energy = 1/2mv2. Potential energy is the mass times gravity times the height. Potential Energy = m*g*h. Conservation of energy states that Ki + Pi = Kf + Pf. This simply means that the sum of the skiers initial potential and kinetic energy is equal to the sum of the skiers final potential and kinetic energy.

With the potential energy at its maximum the skier decides to risk it all and descend. Mirroring the effects of climbing the hill, the skier slowly loses potential energy the farther down the hill the skier is. This causes the skier's kinetic energy to increase, as predicted by conservation of energy. When the skier is at the bottom of the hill, the potential energy has dropped to zero and kinetic energy is once again at its maximum.

Depending on the friction of the snow on the skier, the skier would slow down and come to a rest sometime after reaching the bottom of the hill. The animation below illustrates this concept very well. As the skier goes down the hill, the Kinetic Energy, Potential Energy, Work, and Total Mechanical Energy bars reflect what is happening in the system.

There are some flaws in the ideal scenario above. Cross-country skiers are constantly effected by friction which is not a conservative force. Air resistance and the use of poles are also factors which effect conservation of energy. Despite these outside influences, the animation above still gives a general idea of what happens to work and energy when a skier goes down hill.

Gravity is force which acts on every massive object on earth. Therefore, it only makes sense that it plays into crosscountry skiing as well. Gravity acts on every object on earth with the force equal to mass times gravity. The force of gravity acts on the center of mass.

In order to apply this concept to skiing, image that the picture below is not actually a right triangle and block. Instead pretend the block is a skier and the triangle is actually a nice little hill.

The picture above illustrates that the force of gravity is acting in the downward direction on the skier, with a force of the skier's mass times the acceleration of gravity. The acceleration of gravity is about 9.81 m/s2. However, from the illustration above it is clear that the block-skier is skiing uphill at the angle theta.

In order to calculate the force acting on the skier in the directions parallel and perpendicular to the hill, the mg must be separated into two parts. By separating mg into parts, it becomes clear that the force of gravity pulling a skier down the hill is m*g*sin(theta).

Skiers must consider the force of gravity when choosing equipment. This is because a person's weight and therefore downward pull affects equipment setup. A heavier person might need stiffer skis, whereas a light person may prefer more flexible ones.

Gravity is extremely important in cross country skiing. Without it, going uphill would not be the same. Some people might think skiing uphill would be easier without a force pulling you down the hill. However, gravity also is responsible for keeping you "stuck" to the hill to begin with. Without gravity a skier would be free to float around, and that would probably be the end of skiing right there.

It probably comes as no surprise that acceleration plays a large role in cross country skiing. Rounding a corner, or rather trying to round a corner, while skiing down hill can often lead to disaster. On the other hand, decreasing acceleration (or deceleration) can also cause problems. Imagine a skier accelerating only to be tripped by a hidden tree root. The skier's acceleration would greatly decrease as the ski stopped on the root, and there is a possibility for injury.

One type of acceleration experienced by skiers is linear acceleration. This acceleration is simply the final velocity minus the initial velocity divided by the difference in time. Constant acceleration = (V f -Vi)/(Tf-Ti) . This means that if a skier starts from rest and 2 seconds later the skier is traveling 4 m/s, the skier is accelerating at 2 m/s 2 .

Circular acceleration is another aspect of skiing. There are two components of this acceleration, radial and tangential. Below is a circle with some of the radial and tangential acceleration vectors drawn in.

Newton's first law states that "an object in motion will stay in motion, and an object at rest will stay at rest unless acted upon by an outside force." This means that if there were no outside forces acting on a skier, a single stride would keep the skier moving forward indefinitely. However, there are outside forces which act on skiers. Friction is one example of an outside force. Without friction (and an object in the skier's path) a skier would not slow down after going down a hill.

Works Cited
Beichner, Robert J. Serway, Ramond A. Physics for Scientists and Engineers Fifth Edition. Copyright 2000 Thomson Learning, Inc.

Jacobson, Mark. "The Physics of Skiing." 1998. physics1.html.>

Nave, R. "Friction." August 2000. 20 Nov. 2002.

Rusell, Elizabeth. "Slalom Skiing- AP Physics." 1995-2002 Kent School District. 20 Nov. 2002.

"Work and Energy." 2002 The Physics Classroom and Mathsoft Engineering & Education, Inc. 20 Nov. 2002.
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