Maximum Velocity for Elliptical Orbit

[BBAI BBAI]

Homework Statement

A satellite is at an altitude of 3R (R- Radius of the planet) circulating around the planet with a velocity v. In order to make its path elliptical the velocity is changed from v to βv. What should be maximum value of β such that the satellite doesn't collide with the planet?

Homework Equations

First the expression of V is the orbital velocity and
Initial energy = final energy.( in terms of Gravitation)
I can't remember any other equation.

The Attempt at a Solution

I tried but i didn't get the right answer.. The answer is sqrt(2/5)..Thanks in advance for help..

 

[mfb]

I think you are looking for a minimal value.
Consider the case where the satellite is just "touching" the surface: There are two ways to calculate its velocity, based on two conserved quantities of orbits. They both have to give the same result for a real orbit, this allows to calculate β.

 

[BBAI BBAI]

Please tell your calculation.. i have also taken the consideration of just touching the surface. .

 

[mfb]

Please tell your calculation..

It is your homework problem, not mine. I know how to do it, I don't have to learn that any more.

 

[BBAI BBAI]

Is the answer coming?

 

[mfb]

I helped you to get the answer yourself.

I tried

If you show your work here, I might look for mistakes.

 

[BBAI BBAI]

I have done the following things:
0.5mv^2=GMm/4R.
Then 0.5 Mβ^2V^2-GMm/4R=-GmM/R.
What's the mistake here?

 

[mfb]

0.5mv^2=GMm/4R.

Where does this come from? It is wrong by a constant factor.

0.5 Mβ^2V^2-GMm/4R=-GmM/R.

Why do you have M (instead of m) in the first term? Again, where does the "4" come from? What happened to the velocity at the lowest point of the elliptical orbit?