Question: Finding the closed formula s(t) that gives the approaching position of an inertial mass to a planet Supposing the mass initially stationary, and far enough and for long enough that it is NOT possible to consider the gravity as constant while it moves closer and closer. Said in a different way. Given a mass in free fall from afar to a planet, what is the motion formula s(t), the one that returns: at 0 seconds the mass will be 100000km distant at 10 seconds the mass will be 99999Km distant. at 20 seconds the mass will be 99996Km distant at 100 seconds the mass will be 94000Km distant (it is accelerating, and the acceleration increases while it approaches the planet), etc. So given M= the mass of the planet m = the mass of the free fall mass So = the initial distance of the mass Given F=G(Mm)/r^2 we obtain m*a(t) = GMm/(r(t)^2) a(t) = GM/(r(t)^2) Now we solve with derivative calculus Posing k=GM s'' = k/(s^2) Is the above correct? If it is, how can I solve this differential equation? (I suppose a constant will pop out from some integral, and it will be our initial S0) Can I simply integrate left and right a couple of times? Providing all is preserved, no bad negatives, no 0s around, all functions are analytical in complex space as they appear to be etc. If I can, I suppose that I will end up with s' = S0-k/s s = S0-k/(ln(s)) which is: s(t) = S0-k/(ln(s(t)) And now? How can I get s(t) ? Another way to solve this problem?