A $200 \; kg$. satellite is lifted to an orbit of $2.2\;*\;10^4\;km$ radius. If the radius and mass of the earth are $6.37\;*\;10^6\;m$. and $5.98\;*\;10^{24};kg$ respectively, how much additional potential energy is required to lift the satellite?

 Given,

Mass of the satellite (m) = 200 kg

Radius of the orbit (r) = $2.2 * 10^7\; m$

Radius of the earth (R) = $6.37 * 10^6\;m$

Mass of the earth (M) = $5.98 * 10^{24}\; kg$

Additional potential energy = ?

We have, 

Additional P.E = P.E on the orbit - P.E at the earth's surface

                        = $-\frac{GMm}{r}$ $-$  ($-\frac{GMm}{R})$

                        =  $\frac{GMm}{R} - \frac{GMm}{r}$

                        = $GMm(\frac{1}{R}- \frac{1}{r})$

        
                = $6.7 * 10^-{11} * 5.98 * 10^{24} * 200 \; (\frac{1}{6.37 * 10^{6}} - \frac{1}{2.2 * 10^7})$

                        = $4.47 * 10^7\; J$

$\therefore$ Required Additional P.E. is $4.47 * 10^7\;J$.

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