Finding the (Newtonian) movement equation of an object in a gravitational field

[Username007]

This is a problem I've been looking to solve for some time.

Homework Statement

You must find a movement equation for an object in a gravitational field knowing traditional formulas of force and acceleration of gravity (see below).

Homework Equations

absolute value of the acceleration at a distance 'r' from the centre of gravity of an object with mass 'm'

$g=G\frac{m}{r^{2}}$

'r' as a movement equation

$r=r(t)$
$r''(t)=-g$

The Attempt at a Solution

$r''(t)=-G\frac{m}{r(t)^{2}}$
$r''(t)r(t)^{2}=-Gm$

I don't really know what to do next. It seems that r(t) has at least one constant term and it's of degree>2 if it's a polynomial (as acceleration changes). I know nothing more than that.

 

[JHamm]

It might help to rewrite as
$$\frac{d^2r}{dt^2}r^2 = -Gm$$
After this you should use a relationship you know linking position, velocity and acceleration mathematically.