Dynamics: Get Help Solving Apogee-Perigee Satellite Orbit Problem

[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.

 

[DeeNos]

As a scientist, it is important to approach problems systematically and logically. In this case, you are trying to determine the energy per unit mass required to place a satellite in its elliptic orbit. The given information tells us that the ratio of the velocities at apogee and perigee is equal to the ratio of the distances to the center of the Earth at those points. This means that the satellite will cover the same distance in the same amount of time at both apogee and perigee.

Using this information, we can set up a relationship between the velocities and distances:

Va/Vp = Rp/Ra

Since we are given the distance between apogee and perigee (80,000 km), we can substitute this into the equation:

Va/Vp = 80,000/Ra

Now, we can use the conservation of momentum equation to relate the velocities and distances to the energy per unit mass:

T1 + V1 = T2 + V2
0.5mv1^2 - GMm/Ra = 0.5mv2^2 - GMm/Rp

We can rearrange this equation to solve for the energy per unit mass:

E/m = 0.5v2^2 - GM/Rp - 0.5v1^2 + GM/Ra

Substituting in our relationship between the velocities and distances, we get:

E/m = 0.5v2^2 - GM/Rp - 0.5(v1^2)(Rp/Ra) + GM/Ra

Now, we can use the given ratio again to eliminate the variables:

E/m = 0.5(v2^2)(Ra/Rp) - GM/Rp - 0.5(v1^2)(Rp/Ra) + GM/Ra

Since we are launching the satellite from the surface of the Earth, we can assume that v1 is equal to the escape velocity, which is about 11.2 km/s. We also know that v2 is equal to the velocity at apogee, which we can calculate using the given information:

v2 = (GM/Ra)^0.5

Substituting these values into our equation, we get:

E/m = 0.5[(GM/Ra)^0.5]^2(Ra/Rp) - GM/Rp - 0.5(11.2^2)(Rp/Ra