- #1

zorro

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## Homework Statement

An artificial satellite of mass m of a planet of mass M, revolves in a circular orbit whose radius is n times the radius R of the planet. In the process of motion, the satellite experiences a slight resistance due to cosmic dust. Assuming the resistive force on satellite depends on velocity as F=av

^{2}where a is constant, calculate how long the satellite will stay in orbit before it falls onto the planet's surface.

## The Attempt at a Solution

Click this link

http://latex.codecogs.com/png.latex?av^{2}&space;=&space;m\frac{\partial&space;v}{\partial&space;t}\\\\&space;\int_{0}^{t}adt&space;=&space;\int_{v_{1}}^{v_{2}}&space;\frac{m}{v^{2}}\\\\v_{1}=\sqrt{\frac{GM}{nR}}\\\\v_{2}=&space;\frac{GM}{R}\left&space;(&space;2-\frac{1}{n}&space;\right&space;)\\\\t(incorrect)=&space;\frac{m}{a}\sqrt{\frac{Rn}{GM}}\left&space;(&space;1-\frac{1}{\sqrt{2n-1}}&space;\right&space;)\\\\t(correct)=&space;\frac{m\sqrt{R}\left&space;[&space;\sqrt{n}-1&space;\right&space;]}{a\sqrt{GM}}

I got v

_{2}by conserving energy in the orbit and on the surface of earth.

On substituting these values in my second equation, the answer is coming wrong. Please explain me my mistake.