ET arrives at an alien angle

\(ET\) 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 \(m\ll M_\textrm{spaceship}\) .

Find the angle of projection \(\theta\) for which the package will just graze the surface of Earth.

Details and Assumptions :

  • \(\displaystyle M = \SI{6e24}{\kilo\gram}\)
  • \(\displaystyle R= \SI{6e6}{\meter}\)
  • \(\displaystyle G=\SI[per-mode=symbol]{6.67e-11}{\newton\meter\squared\per\kilo\gram\squared}\)
  • \(\displaystyle\ m = \SI{25}{\kilo\gram}\)
  • Grazing means to make tangential contact.
  • \(v_0 = \SI[per-mode=symbol]{12e3}{\meter\per\second}\)
This question is part of the set Best of Me

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