Take an object a vertical distance $r$ above the surface of the earth of radius $R$.

The acceleration due to gravity at any $r$ is therefore: $a=\frac{GM}{(R+r)^2}$

Lets say the object falls from rest for $t$ seconds, how far does it fall in this time?

I assume calculus is necessary but get lost in the calculations.

$$a=\frac{dv}{dt}$$ so... $$dv=\frac{GM}{(R+r)^2}dt$$

Where do I go from here or am I even on the right track?

I guess I want an integral where: $$ds=\int_0^t something . dt$$

but I don't know how to get there.