Assume that there is no friction as the cars roll along the track. Identify which point or points along the track shown on the diagram (1 through 6) correspond to locations where the first cart of the roller coaster will have a:(a) maximum kinetic energy. Explain.(b) minimum kinetic energy. Explain.(c) maximum potential energy. Explain.(d) minimum potential energy. Explain.(e) zero kinetic energy. Explain

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Three facts about energy will help us solve this problem:

1. Energy is conserved; without any loss to friction, the total energy will remain constant
2. Potential Energy + Kinetic Energy at one point will equal Potential Energy + Kinetic Energy at any other point.
3. PE = mgh and KE =½mv^2

Intuition would tell us that the roller coaster would lose some amount of energy by completing this loop; however, since there is no friction, the total energy is constant. The only thing changing is what proportion of that energy is in potential form or kinetic form.

At Point 1, the "entrance" to the loop, the cart must have all of its energy in kinetic form, and none in potential form; if PE = mgh, and the coaster is at ground level (h=0), then PE = 0. Likewise, at point 6, h=0, so PE = 0 at this point as well. Points 1 and 6 have (A) the maximum kinetic energy and (D) minimum potential energy, because their height is 0.

In contrast, Point 3, at the top of the loop, is as high off the ground as the coaster can get. Therefore it will have the maximum value of h, and therefore the largest potential energy. Point 3 has (C) the maximum potential energy because it has the largest value of h. Since the total energy is constant, and PE + KE = total energy, a large PE will produce the smallest KE value. Point 3 has (B) the minimum kinetic energy because the total energy is constant and potential energy is greatest at this point.

Points 2, 4 and 5 have no special properties; they are all intermediary values. Option E, no kinetic energy, does not appear on this diagram.