# Infinite time for an object to get K=0 conservative Force

#### titansarus

1. The problem statement, all variables and given/known data
Think that we have only conservative force in this question and no nonconservative force like friction exists. Also$U$means potential energy (eg Gravitational Potential Energy),$K$ means Kinetic Energy ($1/2 mv^2$) and $E = K + U = constant$

We have a arbitrary Potential energy (U) in terms of position (x) diagram like the picture. The object is at first, in a position like $x_0$ and wants to get to point $x_1$. At $x_1$ we have $dU / dx = 0$ and Also the initial Mechanical Energy (E = K + U) of the object at first is equal to the U at point $x_1$. We want to prove that it takes infinite time -theoretically- for this object to get to the point $x_1$. (The movement is 1 Dimensional)

2. Relevant Equations
$E = K + U = Constant$
$K = 1/2 m v^2$
$F = m a$
$F = -dU/dx$
Taylor Series at $x = x_1$ for function $f(x)$: $f(x) = f(x_1) + f'(x) (x - x_1) + f''(x) (x - x_1)^2 / 2! + ...$
All forces are conservative in this question.

3. The attempt at a solution

At first we know that the K at $x_1$ equals to zero. Intuitively, I can understand the at the end of the path, the acceleration goes to zero and also the velocity goes to zero (because $K -> 0 ==> V -> 0$). But I don't know how to prove this. I know that F = -dU/dx.

For the start, I get this from taylor series but I don't know what to do with it:

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