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## Homework Statement

Calculate the orbital velocity in a geostationary orbit (the circular orbit around

the Earth which has a period of 24 hours) and show that its radius is approximately

40,000 km.

A satellite is to be inserted into a geostationary orbit from an elliptical orbit with perigee at a geocentric radius of 8,000 km and apogee at 40,000 km. When it is at apogee, a brief firing of its rocket motor places it into the circular orbit. Calculate the change in velocity the motor needs to provide.

## Homework Equations

Conservation of angular momentum and of energy.

L=mr

^{2}(dθ/dt),

**L**=m

**r**x

**v**.

E=m(dr/dt)

^{2}/2+J

^{2}/2mr

^{2}-GmM/r

where M is the mass of the Earth, m that of the satellite.

## The Attempt at a Solution

In the geostationary orbit, I have v=3076ms

^{-1}and R=42000km. Now the first issue is whether or not I can use their approximation of R=40000km and so then assume that the elliptical orbit has the correct radial distance, just not the correct speed at apogee.

Assuming I can do that, equating the energy at apogee and perigee using dr/dt=0 and using r=r

_{min}at perigee and r=r

_{max}at apogee, I can find L

^{2}. Then using L

^{2}=m

^{2}r

_{max}

^{2}v

^{2}at apogee gives v=sqrt{2GMr

_{min}/[r

_{max}(r

_{min}+r

_{max})]}=1826ms

^{-1}.

This concerns me because the energy of an elliptical orbit is greater than that of a circular one. At the position in consideration, they have the same PE, they both have no radial KE contribution, so comparing their tangential KE compares their overall energies. My result that the circular orbit has a greater tangential velocity seems to suggest that the circular orbit has a greater energy. So this must be wrong unless I'm misunderstanding something.

For the sake of completion, the speed boost needed is then an increase of 1250ms

^{-1}. Thanks for any help with these issues in advance.

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