Suppose an object is initially at x = 0 and at rest. It is then acted on by a force F which depends on it’s position as follows
$$ F(x) = +\frac{1}{1+x^2} $$ Without any other forces acting argue why the object never stops. Also, how much energy will the object acquire as x → ∞?
Using Newton's Law, $$m\ddot{x}=\frac{1}{1+x^2}=mv\frac{dv}{dx}$$
So separating the variables and integrating, we have $$\frac 12 mv^2=\arctan x+c$$
From the initial conditions, $c=0$ and as $x\rightarrow\infty$, $\arctan x\rightarrow \frac{\pi}{2}$
Therefore $v$ remains positive and so it continues to move, and the limiting value of the kinetic energy is $\frac{\pi}{2}$