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(A) The total energy of the satellite on the surface of the planet is $E_i = -\frac{GMm}{R}$.The satellite is to be placed in a circular orbit at an a

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$\frac{5GmM}{6R}$

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(A) The total energy of the satellite on the surface of the planet is $E_i = -\frac{GMm}{R}$.The satellite is to be placed in a circular orbit at an altitude of $h = 2R$. The orbital radius is $r = R + h = R + 2R = 3R$.The potential energy of the satellite in the orbit is $U_f = -\frac{GMm}{3R}$.The orbital velocity $v_0$ is given by $v_0 = \sqrt{\frac{GM}{3R}}$.The kinetic energy of the satellite in the orbit is $K_f = \frac{1}{2}mv_0^2 = \frac{1}{2}m\left(\frac{GM}{3R}\right) = \frac{GMm}{6R}$.The total energy in the orbit is $E_f = U_f + K_f = -\frac{GMm}{3R} + \frac{GMm}{6R} = -\frac{GMm}{6R}$.The energy required to launch the satellite is $E = E_f - E_i = -\frac{GMm}{6R} - \left(-\frac{GMm}{R}\right) = \frac{GMm}{R} - \frac{GMm}{6R} = \frac{5GMm}{6R}$.

What is the minimum energy required to launch a satellite of mass $m$ from the surface of a planet of mass $M$ and radius $R$ into a circular orbit at an altitude of $2R$?

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