This is almost not suitable for this forum but I just can't figure it out by myself:

In an exercise I have the following sentence:

> 
> Find the solution the two body problem restricted to a straight line:
> 
> 
> $\ddot x(t)=-\frac{Gm}{x^2(t)},\ x(0)=r\_0>0,\ \dot x(0)=v\_0$
> 
> 
> 

I don't have any problem with the question mathematically, but I was just wondering that from a physical point of view, the acceleration $\ddot x(t)$ that is written in a simplified form that comes from $\ddot x(t)=-\frac{Gmx(t)}{||x(t)||^3} = -\frac{Gmx(t)}{|x(t)|^{3}}=-\text{sign}(x)\frac{Gmx(t)}{x(t)^{3}}=-\text{sign}(x)\frac{Gm}{x(t)^{2}}$

Otherwise the acceleration is always negative, which is not how I imagine the world works.

Or am I wrong?