# 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

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

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