Finding orbital energy per unit mass from apogee-perigee velocity ratio

[kaos4]

Homework Statement

A Satellite is to be placed in an elliptic orbit about the earth. Knowing that the ration Va/Vp of the velocity at the apogee A to the velocity at perigee P is equal to the ration Rp/Ra of the distance to the center of the Earth at P to that at A, and the distance between A and P is 80,000 km, determine the energy per unit mass required to place the satellite in its orbit by launching it from the surface of the earth.

Pic:

Va v-----Ra---------O----Rp----^ Vp

|---------80,000km---------|

Homework Equations

Conservation of Momentum: T1 + V1 = T2 + V2
T = .5mv^2
V = - GMm/r

The Attempt at a Solution

Va/Vp = Rp/Ra
Ra = 80,000 - Rp
E = T + V
E = 0.5mv^2 - GMm/r
E/m = .5v^2 - GM/r

I'm not sure where to go next. I know the final answer is 57.5 MJ/kg. How are all the 'r's and 'v's eliminated by just using the ratio? I'm generally able to solve these questions, but I've been working on this one for hours with no luck. Any help is greatly appreciated!

 

[tiny-tim]

Hi kaos4!

It's launched from the surface of the Earth …

so where is the Earths's radius in your equations? ​

 

[kaos4]

hmm..I considered that. That would give me GM, but still Ra and Rb would be unknown. Unless I am missing something, but the Earth's radius would just add another number, not eliminate any variable (Earths radius is a part of Ra and Rb). Thanks for the advice though, I will try and see how else I can apply the Earth's radius.

 

[kaos4]

http://books.google.com/books?id=A_...s to be placed in an elliptic orbit"&f=false"

It's problem 13.86. The diagram may better explain the problem.