Satellite Planet problem

1. The problem statement, all variables and given/known data

A satellite is moving in a circular orbit around a massive planet of radius R. The altitude of the satellite above the surface of the planet is 3R and its speed is v. In order to place the satellite in an elliptical orbit which will bring it closer to the planet ,its velocity is reduced from v to μv., where μ<1 . The smallest permissible value of μ if the satellite is not to crash on the surface of planet is √(2/K), find K .

2. Relevant equations

3. The attempt at a solution

Orbital speed of the satellite [itex] v = \sqrt{\frac{GM}{4R}}[/itex]

KE of the satellite = [itex]\frac{GMm}{8R}[/itex]

PE of the satellite = [itex]-\frac{GMm}{4R}[/itex]

[itex]E = -\frac{GMm}{8R}[/itex]

Just after the speed is reduced , KE1 = [itex]\frac{1}{2}mμ^2v^2 = \frac{μ^2GMm}{8R}[/itex]

PE1 = [itex]-\frac{GMm}{4R}[/itex]

[itex]E_1 = \frac{μ^2GMm}{8R} -\frac{GMm}{4R}[/itex]

[itex]E_1 = \frac{GMm}{4R}(\frac{μ^2}{2}-1) [/itex]

The semi major axis ‘a’ of the new orbit will be given by [itex]\frac{1}{a} = \frac{1}{2R}(1-\frac{μ^2}{2})[/itex]

Or, [itex] a = \frac{4R}{2 – μ^2}[/itex]

The minimum distance of the satellite from the planet will be a(1-e) where ‘e’ is the eccentricity of the elliptical orbit .

If the satellite is not to crash then a(1-e) > 0 or a>0 . Is this the condition we need to use ?

I am not sure if I have approached the problem correctly .

I would be grateful If someone could help me with the problem.

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