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titansarus

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## Homework Statement

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)

## Homework 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

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:

##U(x) = U(x1) + dU/dx (x - x1) + O(x^2)## -> ##U(x) = U(x1) + (-m a) (x - x1)

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