How do I get part C? Physics help

Question

2) Stairs are about 18 cm in height for each step.

a) If a person of mass 65 kg goes up the steps at a rate of 2 steps per second, what is the change in the person’s gravitational potential energy after 10 seconds of climbing the stairs?

2,293 Joules

b) What is the person’s average power output while climbing the stairs assuming the person’s speed is constant?

In Watts: 229.3 Watts; in horsepower: 0.307 hp.

c) How much energy will it take to lift an object of mass 65 kg from the earth’s surface up a height of 4,200 kilometers above the earth’s surface?

1.62 x 109 Joules

 

How do I get Part C???

Answer 1

Hi Anna!

 

I can see the confusion, because we are so used to calculating gravitational potential energy using:

PE_g = mgh

 

But that assumes that "g" is a constant, which is good if you are not too far from the surface.  But at this altitude, we have moved away from the surface by about 2/3 the average radius of the Earth, which is significant.

 

In this case, the principle is the same:  we have to do an amount of work to raise the body equal to the amount of potential energy the body gains as it is raised.  But, in this case, we have to use the more general form of gravitational potential energy:

 

PE_g = -(GmM/r)

 

where, in this case:

 

m = mass of body

M = mass of the Earth

r = distance from center of Earth to the body

 

So, ΔPE = PE_f - PE_i =[-GmM/(R_E + 4.2x10⁶km)] - [-GmM/(R_E)]) = -GmM([1/(R_E + 4.2x10⁶km)] - [1/(R_E)])

 

If you use standard values for the mass of the Earth and the radius of the Earth, R_E, this should give you the answer you listed (at least, it did for me).

 

Just let me know if you have any other questions about this, or problems!