# A skier wants to try different slopes of the same overall vertical height, h, to see which one would give him the most speed when they reach...

A skier wants to try different slopes of the same overall vertical height, h, to see which one would give him the most speed when they reach the level again (points 1, 2 & 3).

(a) Assuming there is no friction force between the skis and the snow which hill would leave them with the most speed? Which would leave them with the least speed? Explain the basis for your answer.

(b) Assuming there is a noticeable friction force between the skis and the snow which hill would leave them with the most speed? Which would leave them with the least speed? Explain the basis for your answer.

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### 1 Answer

**A.**We know that energy is being conserved according to the equation (PE + KE)i = (PE + KE)f for any two points along the path of motion. The person starts from the same height, and thus he starts with the same potential energy for all three circumstances; PE is the same for all three.

When the skiier reaches the ground, all of that potential energy will be converted into kinetic energy, i.e. velocity.

Points 1, 2 and 3 are all at ground level, where h = 0. Thus, the potential energy at points 1, 2 and 3 must = 0.

If the skiier starts from rest, then the starting kinetic energy is 0.

(PE + 0)i = (0 + KE)f

All of the potential energies are the same. They are all converted completely to kinetic energy. **Thus, if there is no friction, all three hills provide the same final speed.**

**B.**

If there is friction, the conditions change. Friction is a force acting opposite to the skiier's forward motion. The more distance the skiier covers, the more time they spend in contact with the friction-generating surface, and the more that force takes away from their speed. Work and energy are both defined as a Force acting over a distance; if we increase the force or the distance, we increase the work or energy exerted.

The amount of gravity that applies to the normal force changes with proportion to the angle, thus a greater distance is not necessarily a greater friction force.

The loss of energy to friction should be the same for all three points, and their velocity should be the same.