**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.