# 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.
• $$v_0 = \SI[per-mode=symbol]{12e3}{\meter\per\second}$$
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