Gravitation - Satellite in circular orbit

[atavistic]

Homework Statement

A satellite of mass m, initially at rest on the earth, is launched into a circular orbit at a height equal to radius of the earth. What is the the minimum energy required for this purpose?

Homework Equations

GMm/r^2 = mv^2/r

PE at surface = -GMm/R
PE at orbit = -GMm/r

where r = 2R

The Attempt at a Solution

I am not getting the logic, and I think my problem is more so related to circular motion, I mean if we launch a satellite ,then it will go up and if viewed from outside Earth , follow an elliptical trajectory , then how does it get into a circular orbit?How does a force which is actually pulling it downward suddenly provide centripetal acceleration for rotational motion? How is the initial and final energy related. I did most of the problem but this has left me perplexed made me revisit circular motion but I didnt get the solution to my answer anywhere.

 

[DrDan]

To get you on the right track, think of this in terms of energy. You need to do a certain amount of work on the satellite to get it into orbit. That work goes into a change in potential energy - and - you have to get the satellite up to a certain velocity - so you also need to provide enough energy that goes into kinetic energy.

 

[atavistic]

I understand that but that's not really my question but thanks.

 

[nasu]

You are making it more difficult than it is. You don;t have to go into the technical details of the launching. And it does not have to be an elliptical orbit (unless you consider a circular orbit as a special case op ellipse).
You may think about it in two steps:
1. rise the satellite up to a height of 2r - you provide potential energy
2. "kick-it" laterally so it gets the appropriate speed for that orbit - you provide kinetic energy
Total energy provided = sum of the two.
Now, for the orbital motion, you must have mv^2/r = F
F is the centripetal force, here the gravitational force between satellite and Earth. This will give you the orbital speed.