ET arrives at an alien angle

Classical Mechanics Level 5

\(ET\) and his fellow aliens have boarded a spaceship to our planet, Earth of mass \(M\) and radius \(R\). While hanging motionless in space at a distance of \(7R\) from the center of Earth, the spaceship fires with velocity \(v_{0}\), a package in which the aliens are huddled together. Consider the total mass of the package to be \(m\), where \(m<<M_{spaceship}\) .

For a particular angle of projection \(\theta\) the package will just \( graze\) the surface of Earth, which is given by \[{\sin^{-1}(\dfrac {\sqrt p}{q})}\]. Find \(p+q\)

Details and Assumptions :

\(\bullet\) \(\displaystyle M = 6\times 10^{24} kg\)

\(\bullet\) \(\displaystyle R= 6400 km\)

\(\bullet\) \(\displaystyle G=6.67\times 10^{-11}Nm^{2}kg^{-2} \)

\(\bullet\) \(\displaystyle\ m= 25 kg\)

\(\bullet\) \('Grazing'\) means to brush past / to have a light contact

\(\bullet\) \(\displaystyle\ v_{0}=12km/s\)

This question is part of the set Best of Me

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