Energy transfer from x altitude to geosynchrous altitlude

[tnutty]

Homework Statement

INTRO TO THE PROBLEM :For a circular orbit around a massive gravitating body, the speed depends on the radius according to the equation V = sqrt (GM/r) ; for elliptical orbits, the speed varies according to the equation v^2 = 2GM([ 1/r - 1/(2a), where r is the distance from the massive body and a is the semimajor axis of the ellipse (i.e., half the sum of the closest and farthest distances). A satellite can be transferred from one circular orbit (at radius r1) to a higher orbit (at radius r1) by boosting the circular speed v1 at v2 to the appropriate speed for an elliptical orbit whose distance varies between r1 and r2 , and then boosting the speed in the elliptical orbit at r2 to the circular speed v2. This is called a Hohmann transfer.

THE PROBLEM STATEMENT :
How much energy is required for the first boost in such a transfer to take a 280kg satellite from a circular orbit at a 400 km altitude to the altitude of a geosynchronous orbit?

change in K_1 = _________J

Homework Equations

stated above in the intro

The Attempt at a Solution

clueless?

 

[LowlyPion]

The conservation of energy still works doesn't it?

(Potential + Kinetic) before + Energy = (Potential + Kinetic) after?

 

[tnutty]

so,

1/2mv^2 - GMm/r = 1/2mv^2 - GMm/r

?

 

[LowlyPion]

so,

1/2mv^2 - GMm/r = 1/2mv^2 - GMm/r

?

The Δ anyway. Mustn't that be the additional energy?

 

[tnutty]

I'm sorry i am confused

 

[LowlyPion]

I'm sorry i am confused

You have initial potential energy and kinetic energy in lower orbit.

You add energy.

You get a higher orbit with kinetic energy and potential energy.

 

[tnutty]

how about saying it mathematically. can you start me off?

 

[LowlyPion]

how about saying it mathematically. can you start me off?

You have the equation below. I'm sure you can do it.

Besides if it comes up on a quiz, I won't be there to start you off. It's good practice.

 

[tnutty]

So what's the formula to delta k ?

 

[LowlyPion]

So what's the formula to delta k ?

Don't they give you the formulas for how to calculate the v²'s

 

[tnutty]

ok. r is the radius of the Earth + 400km and a is ? And that formula above v^2 =...
is delta K ?

 

[LowlyPion]

ok. r is the radius of the Earth + 400km and a is ? And that formula above v^2 =...
is delta K ?

Well that's a good part of it, but strictly speaking you want to be working with ½mv² for KE. so use the v² 's that you find to determine your ½mv² 's.

 

[tnutty]

How would I represent (a) as? in the equation above?

 

[LowlyPion]

Don't look now, but they tell you how to determine a in the problem. If your ellipse is to vary between r1 and r2, what is the semi-major axis?

 

[tnutty]

(r1+r2) / 2

 

[LowlyPion]

(r1+r2) / 2

That would seem so.

So plug 'em in and get'er done.