Consider the diagram, a probe of mass m is projected with a velocity of $v_{0} = \sqrt{\frac{G M}{4 R}}$ in tangential direction. The probe has a rocket which fires to put it into a circular orbit at that height. Assume that, at all times, the probe is at the same distance from the planet and that there are no forces acting on the probe other than the planet’s gravity and the rocket. The work done by the rocket on probe is found to be $\frac{G M m}{3 n R}$ . Find n

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answer is 8.

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Detailed Solution

For a circular orbit at distance 3R
$m \frac{v^{2}}{3 R} = \frac{G M m}{{(3 R)}^{2}} \Rightarrow v = \sqrt{\frac{G M}{3 R}}$
$W_{r o c k e t} = Δ E, Δ P E = 0$ , because always at the same distance.
$W_{r o c k e t} = K E_{F} - K E_{r}$

$= \frac{1}{3} m v_{F}^{2} - \frac{1}{2} m v_{1}^{2} = \frac{1}{2} m \frac{G M}{3 R} - \frac{1}{2} \frac{G M}{4 R} = \frac{G M m}{R} (\frac{1}{6} - \frac{1}{8})$