Why will this object never stop?

Question

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 → ∞?

Notice that $\vec F=m\vec a$ if $m$ doesn't change over time. For the energy, recall that the kinetic energy of the object $E_k$ is solely provided by the work of $F$, and that $W=\int \vec F\cdot d\vec s$. — Frenzy Li, Apr 19 '17 at 06:39

score 0 · Answer 1 · answered Apr 19 '17 at 08:30

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 $\frac{\pi}{2}$

Why will this object never stop?

1 Answers1