Homework Help: Gravitational potential energy: derive expression for energy

1. The problem statement, all variables and given/known data
Part A:
Derive an expression for the energy needed to launch an object from the surface of Earth to a height h above the surface.

Part B:
Ignoring Earth's rotation, how much energy is needed to get the same object into orbit at height h?

Express your answer in terms of some or all of the variables h, mass of the object m, mass of Earth mE, its radius RE, and gravitational constant G.


2. Relevant equations
U=-GmmE/RE+h

3. The attempt at a solution
I figured that the energy you would need to bring the object to height h above the surface would equal the difference between the energy at the surface and the energy at h. I assume the object is at rest and the only energy it has is gravitational potential.

E=-GmmE/RE-(-GmmE/RE+h)....E=GmmE/h

This is not correct. Feedback I got was that the answer depends on RE. I've no idea how to go about part B.

 

E=(-GmmE/Re)+(GmmE/Re)+(GmmE/h) First two terms cancel? I may have forgotten basic algebra.

 

E=(-GmmE+h/Re+h)+(GmmE/Re+h)

Can't believe I did that. Can I add +h to the numerator and denominator of the first term? Is that finding a common denominator? Then I get E=h/(Re+h).

 

Reworked and got E= [-GmmE(Re+h)+Re(GmmE)]/Re(Re+h)=GmmEh/Re(Re+h)

Doing the algebra correctly, is the answer right? Is my reasoning sound? Sorry for forgetting elementary school, and thanks for your help.

 

Very helpful. I got to the right answer. Thank you!