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**Hi all,**

This question is for an assignment due Friday. My friend and I spent the better part of an hour on it and are thoroughly stumped. Any help is much appreciated!

1. Homework Statement

This question is for an assignment due Friday. My friend and I spent the better part of an hour on it and are thoroughly stumped. Any help is much appreciated!

1. Homework Statement

Calculate the energy required to put a satellite into geostationary orbit. (note that no mass is provided)

## Homework Equations

Derivation:

PE

_{gravitational}+ KE = E

_{total}

-GMm/r + 1/2mv

^{2}= E

_{total}

Note that when in non-inertial reference frames,

Fg = mv

^{2}/r

GmM/r

^{2}= mv

^{2}/r

GM/r

^{2}= v

^{2}/r

GM/r = v

^{2}

Plug this value in place of v

^{2}

-GMm/r + 1/2m(GM/r) = E

_{total}

-1/2GMm/r = E

_{total}

So the energy of the satellite once in stable orbit is -1/2GMm/r, or one half of it's potential energy.

## The Attempt at a Solution

Calculate altitude of geosynchronous orbit. At geosynchronous orbit, only force on the satellite is gravity.

Fnet = ma

Fnet = GMm/r

^{2}

ma = GMm/r

^{2}

mv

^{2}/r = GMm/r

^{2}

V = dist/ time

V = 2rpi / T

m(2rpi/T)

^{2}/r = GMm/r

^{2}

m4rpi

^{2}/T

^{2}= GMm/r

^{2}

4rpi

^{2}/T

^{2}= GM/r

^{2}

Isolate for r:

4r

^{3}pi

^{2}/T

^{2}= GM

r = 3√(GMT

^{2}/4pi

^{2})

= 42 250 474m

Subtract radius of Earth to determine height of geosyncronous orbit above surface:

42 250 474metres - 6 371 000metres = 35 879 474metres

Now it should just be a matter of plugging numbers into the equation for 1/2PE

_{g}. Since I don't know the satellites mass I'll represent it with 'm' as a placeholder:

1/2PE

_{g}= -GMm/2r

= -(6.67*10

^{-11}* 5.98 * 10

^{24})m / 2(35 879 474)

= -5 558 414m (in juoles)

Note again that m is a placeholder for the satellite's mass.

While mathematically I can't find an error, it seems intuitively wrong that it requires negative energy, if that even exists, to launch a satellite into geosynchronous orbit. It takes a lot of energy to get it moving sideways so fast, and even though the potential energy does decrease with the rise in altitude it just seems so wrong that the whole process involves negative energy.

Can somebody break this down for me, pointing out either a) how it's possible to have negative energy required to launch a satellite or b) where I've gone conceptually or mathematically astray?

Thanks!