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Tanya Sharma
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Homework Statement
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.
Homework Equations
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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