Satellite bullet problem

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

An earth satellite is revolving in a circular orbit of radius ‘a’ with velocity ‘v0‘. A gun is in the satellite and is aimed directly towards the earth.A bullet is fired from the gun with muzzle velocity v0/2.Neglecting resistance offered by cosmic dust and recoil of gun,calculate maximum and minimum distance of bullet from the center of earth during its subsequent motion.

2. Relevant equations

3. The attempt at a solution

Orbital speed of satellite is [itex]\sqrt{\frac{GM}{a}}[/itex]

Initial velocity of the bullet [itex]v_{i} = \sqrt{{v_o}^2+(\frac{v_0}{2})^2} = \frac{\sqrt{5}v_{0}}{2}[/itex]

Let P be the point at which bullet is fired and Q be point where distance is maximum/minimum.

Applying conservation of angular momentum at P and Q


or , [itex]v = \frac{v_{i}a}{r} = \frac{\sqrt{5}}{2}\frac{av_0}{r}[/itex]

Applying conservation of mechanical energy at P and Q

[itex]\frac{1}{2}m{v_i}^2 – \frac{GMm}{a} = \frac{1}{2}m{v}^2 – \frac{GMm}{r}[/itex]

Solving the equations , I get [itex]3r^2-8ar+5a^2 = 0 [/itex] which gives r =5/3a and a .

The answer i am getting is incorrect .

The correct answer given is 2a and 2a/3 .

I would be grateful if somebody could help me with the problem.

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