# Interpreting a Potential Energy Graph

[SOLVED] Interpreting a Potential Energy Graph

## Homework Statement

A particle moving in one dimension is subject to the potential energy curve shown in
the plot. Which of the following statements is true?

[Diagram attached]

A) At point xa, the force on the particle is in the negative x direction.
B) If the particle has total energy of 3 Joules at point xc and its velocity is in the positive
x direction, it will eventually reach point xa.
C) If the particle has kinetic energy of 3 Joules at point xc it can reach point xb.
D) If the total energy of the particle at point xb is 3 Joules it can reach point xc.
E) If the particle is released from rest at xc, it will travel in the positive x direction.

F = -dU/dx

## The Attempt at a Solution

I know A is false because F is the negative derivative of U. The slope at point Xa is negative, so when I take the negative of that, F ends up being positive.

Other than that, the wording confuses me. I'm sure that it's a whole lot simpler than I'm making it out to be.

I'm not just looking for the answer (I know it's C), I'd like to understand why the other choices are false.

#### Attachments

• PEgraph.jpg
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## Answers and Replies

Hi Moxy

I think you could interpret the graph a lot simpler. As the potential energy is plottet agains the direction x, you can simply think about the whole thing as kind of a "hill" (or several "hills" in this case). If you look at the statements, it should work:

A) If you would release the particle at x_a, it would roll down the 'hill' in the positive x direction -> the force is in the positive direction. (Your approach with F=-dU/dx works perfectly here too)

B)The potential Energy of the particle at the point x_c is 1 Joule, with the given total energy of 3 Joules the kinetic Energy has to be 2 Joules (assuming there's no friction). As the graph goes higher than 3 Joules between x_b and x_c, the particle cannot reach x_b or x_a.

C)Potential energy + kinetic energy = 4 Joules -> particle can reach x_b

D)+E) should be clear with this...

I don't know if this helps... Sorry for my poor English

B) is false. At point xc, the potential energy measures up to 1J, and if the total energy measures up to 3J, then 2J goes to kinetic energy. Due to the conservation of energy, the particle cannot reach point xa; between points xc and xb, there is a hill that measures beyond 3J...to get past that point, it needs at least that amount of energy, which it does not.

D) is false for the same reason B is false; there is a hill between points xb and xc, in which more than 3J is needed.

E) is false because F = -dU/dx, in which the force there will point in the negative x-direction (and from rest, the particle has no initial velocity).

C) is the correct choice because at point xc, there is 1J of potential energy and 3J of kinetic energy. The total energy measures up to 4J, which is more than the amount needed to surpass the hill between xc and xb.

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Okay, those both make sense. I kind of figured the "peak" in the graph between Xb and Xc had something to do with it, but I wasn't sure exactly what it implied. I figured the question was a lot simpler than I was making it. The hill analogy made it much easier to conceptualize.

Thanks!