There are motions which are difficult to predict using just the concept of force and the basic forms of Newton’s Laws. For example it is possible but very tedious to predict the motion of a...

There are motions which are difficult to predict using just the concept of force and the basic forms of Newton’s Laws. For example it is possible but very tedious to predict the motion of a string of carts on a low friction roller coaster by calculating the net force on the carts at each point along the coaster tracks. This is because as the slope of the track changes the components of the gravitational force, the normal force and the friction force also change. The concepts of work, kinetic energy, and potential energy were invented by physicists. Then they used Newton’s Second Law in a mathematical derivation that leads to the work-energy theorem. This W-E theorem, in turn, makes the law of conservation of energy plausible.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 he/she reaches the level again (points 1, 2 & 3). His/her options are shown in the diagram to the left.

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

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caledon | High School Teacher | (Level 3) Senior Educator

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The conservation of energy tells us that, in the absence of friction, the total amount of energy in the system will remain constant. Since the skier begins each trip at the same height, H, and ends at the same height (the ground plane), the amount of energy in these systems is always the same.

PE(initial) + KE(initial) = PE(final) + KE(final)

The KE(initial) on each hill is 0. The PE(final) on each ground plane is 0.

So:

PE(initial) + 0 = 0 + KE(final)

PE(initial) = KE(final)

PE = mgh and KE = 1/2mv^2

mgh = 1/2mv2

gh = .5v^2

√19.6h = v for each Point.

If we include friction, we know that the final speed will be reduced, because friction is working against the forward motion created by gravity and the angle of the hill. The problem, as described in the introduction to this question, is that the angle of the hill changes, altering the normal force at numerous points, and thereby altering the portion of gravity that contributes to friction via the normal force.

However, this is not as difficult as it seems. The most direct path from the top of the hill to the Points on the ground would be a straight line. If we know the force of friction, we could calculate its action over this distance via W=Fd, thereby finding the energy that friction takes away from the skier.

While it would intuitively appear that a steeper hill would reduce friction, and result in a faster skier at the end, this is not the case. True, the steeper the hill, the less normal force, and the less friction, but a steep hill results in a greater shallow-angled distance to cover to reach one of the points, and shallow angles have a large normal force. Thus, any modification of the skier's path will be balanced by other parts of the path, resulting in the same velocity at each point.

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