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(A) For a satellite in a circular orbit of radius $r$ around the Earth of mass $M$,the gravitational potential energy is $U = -\frac{GMm}{r}$ and the

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$E_k$

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(A) For a satellite in a circular orbit of radius $r$ around the Earth of mass $M$,the gravitational potential energy is $U = -\frac{GMm}{r}$ and the kinetic energy is $E_k = \frac{GMm}{2r}$.The total mechanical energy of the satellite is $E = E_k + U = \frac{GMm}{2r} - \frac{GMm}{r} = -\frac{GMm}{2r} = -E_k$.To escape into outer space,the total energy of the satellite must become at least $0$.Let the additional kinetic energy required be $\Delta E$. Then,$E + \Delta E = 0$.Substituting the value of $E$,we get $-E_k + \Delta E = 0$,which implies $\Delta E = E_k$.Therefore,an additional kinetic energy equal to $E_k$ must be provided to the satellite.

$A$ satellite with kinetic energy $E_k$ is revolving around the Earth in a circular orbit. How much more kinetic energy should be given to it so that it may just escape into outer space?

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