What is the minimal velocity boost to bring a satellite down to Earth?

[2DGamer]

A satellite is orbiting Earth in a circular orbit of radius R. At some point the satellite has to be returned to Earth and so it is given a sudden negative velocity boost in the direction opposite the satellite's forward velocity). The radius of the Earth is r, and the gravitational acceleration at the surface of the Earth is g. Find the minimal boost required to bring the satellite down. (neglect air resistance).


Ok, this seems pretty straight forward and not too difficult, however it's proving to be quite difficult. I know that initially the satellite is moving in a circular orbit so the eccentricity is zero. But when it slows down the orbit becomes elliptical with the perigee on the surface of the earth. Any help whatsoever will be greatly appreciated! Thanks!

 

[JaredJames]

Do you know radius R?

If so, you can then work out what it's orbital velocity is.

And from there, you can then know how much deceleration is required to drop it out of orbit.

That's a fairly simplistic view.

 

[2DGamer]

No, I don't know what R is. Just that it's the orbital radius of the satellite. I don't think I'll get an actual number here. Probably an equation that will allow me to plug in any value for R.

 

[JaredJames]

Well this sounds like homework so I can't just give you the answer.

As I laid out above, follow that and you will arrive at an answer which would be:

Orbital Velocity must be < Xm/s to drop out of orbit and so an acceleration of at least -Xm/s² is required must be applied to the orbiting satellite.

The difference is instead of working with numbers, you are simply rearranging the values to give an equation for orbital velocity and from there you know how much to slow it down by.