Conservation of energy problem?

Question

http://img221.imageshack.us/img221/8995/fzksmg3.jpg

Answer 1

1- Why does the skier loses contact at the crest?
Because speed v is greater than a certain speed.

What is this certain speed?
Since v²/R and g are both acceleration at top of the crest,
v²/R = g gives you this speed (g is gravity).

2 - Now you have v, you want to get h.
Suppose Potential energy is entirely transformed into kinetic energy (conservation of energy):
mgh = 1/2 mv²
h = 1/2 v²/g = 1/2 R
h has to be greater than 1/2 R

Answer 2

1. The height "h" of the hill will determine the skier's speed at the crest of the circle (this can be figured from conservation of energy).

2. The skier's speed at the crest of the circle, will determine whether she loses contact. This can be figured by using the formula for acceleration for circular motion (a=v²/r), and comparing her acceleration to the forces on her, using F=ma.

Part 1. By conservation of energy, her total energy at the start (at top of high hill) must equal her total energy at the circle crest:
KE_initial + PE_initial = KE_final + PE_final
0 + mgh = mv²/2 + 0

You can solve that equation to get "v" in terms of "g" and "h".

Part 2: During most of the trip, the skier has 2 forces acting on her: (1) gravity; and (2) the "normal force" of the snow pushing up on the skis. But if she "just loses contact" at the crest of the circle, it means that the normal force has dropped to zero (snow can't push on skis if they're not touching).

Therefore, the only force acting on her, at the crest of the circle, would be her weight (mg):

F_net = mg [when "losing contact" at circle crest]

By F_net=ma, her acceleration at that point is:

a = F_net/m = (mg) / m = g

Now, whenever something travels in a circle with speed "v", its acceleration toward the circle's center is:

a = v²/r

So, since we showed that a=g, we have:

g = v²/r

Now combine this with the equation for "v" that you got in Part 1; then solve for "h" in terms of "r".