Calculating the distance crossed by a ball under changing gravitational acceleration

Question

Suppose a ball of finite mass is taken to such a height in space that the gravitational acceleration decreases significantly. Now, as you let go of the ball, it should head straight down towards the surface of earth, and as it crosses distance and gets closer to the earth surface, it's acceleration should increase. How would I, in this case, be able to calculate the distance traveled by the ball in a certain time?

Answer 1

$$\begin{align}F&=-\frac {GMm}{x^2}\\ a&=-\frac {GM}{x^2}\\ v\frac {dv}{dx}&=-\frac {GM}{x^2}\\ \int v\;{dv}&=\int-\frac {GM}{x^2}\;dx\\ \frac {v^2}2&=\frac {GM}x\\ v&=\sqrt\frac {2GM}x\\ \int\sqrt x\;dx&=\int\sqrt {2GM}\;dt\\ \frac 23x^{\frac 32}&=t\sqrt {2GM} \end{align}$$

Here, you have an equation relating distance and time in a gravitational field.