#$&*
course Phy 201
The gravitational PE of the skier decreases by 60 kg * 9.8 m/s^2 * 65 m = 38 kJ, approx. (this means 38 kiloJoules, or 38,000 Joules). The PE loss partially dissipated against friction, with the rest converted to KE, resulting in KE = 38 kJ / 10.5 kJ = 27.5 kJ.
Formally we have
`dKE + `dPE + `dWnoncons = 0, where `dWnoncons is the work done by the skier against friction. Since friction does -10.5 kJ of work on the skier, the skier does 10.5 kJ of work against friction and we have `dKE = -`dPE - `dWnoncons = - (-38 kJ) - 10.5 kJ = 27.5 kJ.
The speed of the skier at this point will be
v = sqrt( 2 KE / m) = sqrt( 2 * 27,500 J / (65 kg) ) = 30 m/s, approx.
Over the 82 m patch the force exerted against friction will be .2 * 60 kg * 9.8 m/s^2 = 118 N, approx., so the force exerted against nonconservative forces will be 118 N + 160 N = 280 N approx.. The work done will therefore be
`dWnoncons = 280 N * 82 m = 23 kJ, approx.,
and the skier's KE will be
KE = 27.5 kJ - 23 kJ = 4.5 kJ, approx.
This implies a speed of
v = sqrt( 2 KE / m) = 12 m/s, approx.
To stop from this speed in 2.5 m requires that the remaining 4.5 kJ of KE be dissipated in the 2.5 m distance. Thus we have
`dW = Fave * `ds, so that
Fave = `dW / `ds = 4500 J / (2.5 m) = 1800 N.
This is a significant force, about 3 times the weight of the skier, but distributed over a large area of her body will cause a good jolt, but will not be likely to cause injury.**
STUDENT QUESTION
If the PE is = 20.6, then why is the initial KE= 20.6 so that we are adding the .245 to the initial KE of 20.6 to get
20.8, I thought that the KE was equal and opposite to the PE why would we not subtract here?
INSTRUCTOR RESPONSE
`dKE + `dPE = 0, provided there are no nonconservative forces acting on the system.
In such a case, PE goes up as the mass rises, so KE goes down. Another way of looking at it: All or part of the KE converts to PE.
The mass can only go as high as the initial KE permits. Once the initial KE is 'used up', no increase in PE is possible (recall the assumption that no nonconservative forces act during this phase of motion).
At maximum height the mass is still moving in the horizontal direction, so not all of the KE converts to PE.
In this case PE increases by 20.6 M m^2/s^2, .245 M m^2/s^2 of KE is still present at the highest point, so about 20.8 M m^2 / s^2 of KE must have been present initially.
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