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A satellite is revolving around the earth with a kinetic energy $E$. The minimum addition of kinetic energy needed to make it escape from its orbit is
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$E$
(d): Orbital velocity of satellite, $v_0=\sqrt{\frac{G M}{r}}$
Kinetic energy, $E=\frac{1}{2} m v_0^2=\frac{1}{2} \frac{G M m}{r}$
Kinetic energy for escaping,
$\begin{aligned} E^{\prime}=\frac{1}{2} m v_e^2 & =\frac{1}{2} m \frac{2 G M}{r}=\frac{G M m}{r} \\ & =2 E \quad\left(\because v_e=\sqrt{\frac{2 G M}{r}}\right)\end{aligned}$
Additional kinetic energy needed $=E^{\prime}-E=2 E-E=E$
Kinetic energy, $E=\frac{1}{2} m v_0^2=\frac{1}{2} \frac{G M m}{r}$
Kinetic energy for escaping,
$\begin{aligned} E^{\prime}=\frac{1}{2} m v_e^2 & =\frac{1}{2} m \frac{2 G M}{r}=\frac{G M m}{r} \\ & =2 E \quad\left(\because v_e=\sqrt{\frac{2 G M}{r}}\right)\end{aligned}$
Additional kinetic energy needed $=E^{\prime}-E=2 E-E=E$
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