Potential Energy?

Question

A skier starts down a frictionless 30{^\circ} slope. After a vertical drop of 27m, the slope temporarily levels out and then drops at 22{^\circ} an additional 41m vertically before leveling out again.

A. What is the skier's speed on the upper level stretch?

B. What is the skier's speed on the lower level stretch?

My Work for A:

Initial: K + U = Final: K + U
K = kinetic energy, U= potential energy
Initial: 1/2 mv^2 = mgh = Final: 1/2 mv^2
Since it's a 30 degree slope, force of gravity is:
X direction: mgsin30
Y direction: mgcos30
Where gravity = 9.8 m/s^2
Height: 27m

Before the 27m drop, does the skier have kinetic energy? I would think so because he's moving but does he have potential energy right before he drops? If so, does the kinetic+potential before he drops have to equal to 1/2 mv^2 once he is on the stretch?

I am confused.

Thanks.

Answer 1

I would assume the skier starts from rest and would define a height of zero to be the height at the lower level stretch. This would mean that the skier starts with a height of 69 m and the upper level stretch is at a height of 41 m. The nice thing about working in terms of energy rather than forces is the fact that the angle of the slope is irrelevant as potential energy does not depend on the angle of your slope.


I will define the following quantities for use in both parts.

m = skier's mass
h0 = 69 m =height when the skier starts
h1 = 41 m =height of upper stretch
h2 = 0 m = height of lower stretch
v0 = 0 m/s = velocity when skier starts
v1 and v2 are the velocities at the upper and lower stretch respectively.
g = acceleration of gravity

For part A.

Initial K = 0 J
Initial U = m g h0
Final K = 1/2 m v1^2
Final U = m g h1

m g h0 + 0 = 1/2 m v1^2 + m g h1

mass will cancel out leaving us with:

g h0 = 1/2 v1^2 + g h1
g (h0 - h1) = 1/2 v1^2
(2g (h0 - h1))^(1/2) = v1
(2 * 9.8m/s/s * (69 m - 41 m))^(1/2) = v1
(I don't have a calculator handy, but would assume you could figure it out from here).

Part B.

We can use a very similar procedure, since energy is conserved.