6. A 55 kg skier is at the top of a slope, as shown in the illustration below. At the initial point A, the skier is 10.0 m vertically above the final point B.
a. Set the zero level for gravitational potential energy at B, and find the gravitational
potential energy associated with the skier at A and at B. Then find the difference
in potential energy between these two points.
b. Repeat this problem with the zero level at point A
c. Repeat this problem with the zero level midway down the slope, at a height of 5.0
m.

The question mentions an illustration that is not provided, but I can solve the problem anyway.

Ua - Ub = 5,390 J regardless of the reference height

Explanation:

Gravitational Potential Energy

Gravitational potential energy is the energy stored in an object because of its vertical position or height in a gravitational field.

It can be calculated with the equation:

U=m.g.h

Where m is the mass of the object, h is the height with respect to a fixed reference, and g is the acceleration of gravity or .

The problem is about proving the difference in potential energy does not depend on the reference used to calculate them.

a. Let's set the zero level for gravitational potential energy at B, i.e., h=0 at B.

The m=55 kg skier is at the top of the slope (point A) at a height of h=10 m.

The gravitational potential energy Ua is:

Ua = 55 * 9.8 * 10 = 5,390 J

The gravitational potential energy Ub (h=0) is:

Ub = 55 * 9.8 * 0 = 0 J

The difference is:

Ua - Ub = 5,390 J - 0 J = 5,390 J

b. Now fix the zero level at point A (h=0).

The gravitational potential energy Ua (h=0) is:

Ua = 55 * 9.8 * 0 = 0 J

Point B is at a height h=-10 m since it's below the reference.

The gravitational potential energy Ub (h=-10) is:

Ub = 55 * 9.8 * (-10) = -5,390 J

The difference is:

Ua - Ub = 0 J - (-5,390 J) = 5,390 J

We get the same value.

c. Finally, fix the reference at the midpoint between A and B.

Point A is at a height h=5 m since it's 5 m above the reference.

The gravitational potential energy Ua is:

Ua = 55 * 9.8 * 5 = 2695 J

Point B is at a height h=-5 m since it's 5 m below the reference.

The gravitational potential energy Ub is:

Ub = 55 * 9.8 * (-5) = -2695 J

The difference is:

Ua - Ub = 2695 J - (-2695 J) = 5,390 J

Again, we get the same value.

Conclusion:

Ua - Ub = 5,390 J regardless of the reference height

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