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Consider an isolated system consisting of a planet with mass $M$ and radius $R$ some distance $x$ away from an asteroid with radius $r$ and mass $m$. The two bodies are initially at rest, and will start falling towards each other due to gravitational attraction.

In terms of the given variables and the universal gravitation constant $G$, what is the speed the asteroid will have when it reaches the planet's surface?

Note: The distance $x$ between them initially represents the distance between their centers of mass. Take both bodies to be uniform and spherical.

A

$\sqrt{2gx}$

B

$\sqrt {\cfrac {2GM^{2} (\frac 1 x - \frac {1} {R+r})} {M + m}}$

C

$\sqrt{\cfrac{2GM}{(x-R-r)}}$

D

$\sqrt {2GM (\frac 1 x - \frac {1} {R+r})}$

E

$\sqrt{\cfrac{2GM} {x}}$

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