Ratio between change in Energy(Gravitational Potential)

[fahlim003]

Ratio between change of Energy(Gravitational Potential)

I would like some assistance in understanding the ratio of change in energy between two objects.

In this case, if object A goes from rest to an altitude (h) above Earth but does not begin an orbit versus object B goes from rest to the same altitude and then proceeds to orbit the earth.

The variables are as follows:
Re=radius of earth
h=altitude
G=universal gravitational constant
M=mass of earth
m=mass of object(A=B, also known as A and B are the same)

I'll need to transcribe my findings, which until now when comparing to my professors simplication are slightly off, a little later.

I was told my error might occur in the initial change of energies for the respective objects, and this may very well be true. However, I did come across great challenges at first when trying to find a GCF for the variable laden energies.

What I've been using prior to simplication:
Object A = change in Eg = Eg2 - Eg1
Object B = change in total energy = Ek - Eg1

Eg as known is a negative value while Ek is positive.

Any help is appreciated and I understand little help may be offered since I've yet to put up the actual attempts, but when I have more time I'll transcribe my most recent effort into mspaint.

 

[fahlim003]

A good turnout.
Ok, here's my most recent work.

http://img410.imageshack.us/img410/3044/ratioworkingqk8.png


Am I not correct in choosing the E2 and E1 for the object heading into orbit?
I choose Total Orbital Energy (-Ek) since that is energy required to orbit for object B. Although, wouldn't it be possible to put change of energy for the object heading to altitude h as simply Ek instead of -Ek-(Eg)? I would really like some input on this.


Also, it's been a long time since I did any Calculus and or math in general, which may explain why I have trouble with simplification.

Some ratios, albeit told as wrong by my professor include:

http://img296.imageshack.us/img296/3913/resultseh6.png

With the last in the line being the earliest example and least correct.

As mentioned, the first two were said to be nearly correct, although it's been a few days since I spoke to her so I cannot say which is more correct.

Again, help is appreciated. Thank you.