Suppose an object of mass $m$ starts at rest at a radial distance $ r_0$ from a perfectly spherical mass $M$ (where $m << M$), $r_0 > R =$ radius of $M$.

Can we analytically determine when $m$ will hit the surface of the $M$?

In other words, can we analytically solve this initial value problem:

$$ \frac{d^2r}{dt^2} ~=~ - \frac{GM}{r^2} ,$$ $$ \dot{r}(0) ~=~ 0 ,$$ $$ r(0) ~=~ r_0? $$


I believe that is covered by this answer I posted some time ago to a related (but not quite the same) question. Adapting it to your notation,

$$t = \frac{1}{\sqrt{2G(m + M)}}\biggl(\sqrt{r_0 R(r_0 - R)} + r_0^{3/2}\cos^{-1}\sqrt{\frac{R}{r_0}}\biggr)$$

The same formula is given in the Wikipedia article Qmechanic mentioned in a comment.


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