# 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 ‘v_{0}‘. A gun is in the satellite and is aimed directly towards the earth.A bullet is fired from the gun with muzzle velocity v_{0}/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

[itex]mv_{i}a=mvr[/itex]

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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