The law of universal gravitation states: $$\vec{F}=-\frac{GMm}{r^2}\hat{r}$$ where we have the familiar $m$ for the small mass and $M$ for the big mass, $r$ being the distance between the center of masses of the two bodies. We can set this equation equal $mg$ where $g$ is the gravitational acceleration - we then have: $$\vec{g}=-\frac{GM}{r^2}\hat{r}$$ Assume an object which is very far from the big mass, and also assume that that object has relatively small mass compared to the big mass (i.e. $\ll M$), so that it does not produce strong gravitational field that would move the big mass. We then have this differential equation: $$\ddot{r}=-\frac{GM}{r^2}$$ What then would be the solution to this equation? My attempt looked like this: $$\frac{d^2r}{dt^2}=-\frac{GM}{r^2}$$ We can use the chain rule, $$\frac{d}{dt}=\frac{d}{dr}\frac{dr}{dt} \Longrightarrow \frac{d^2r}{dt^2}=\frac{d}{dr}\left(\frac{dr}{dt}\right)^2$$ which is then, $$\frac{d}{dr}\left(\frac{dr}{dt}\right)^2=-\frac{GM}{r^2}$$ Then integrating the expression gives us: $$\left(\frac{dr}{dt}\right)^2=\frac{GM}{r}+\mathrm{const}.$$ And I cannot solve this. I am not even sure if my method of solving this is correct, can someone help me or give me a link to a solution?
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