# Approximating the time it takes for a particle with a potential $-Ax^4$ to approach the origin [closed]

Here's the problem I'm solving:

A particle of mass $$m$$ can only move along the $$x$$-axis and is subject to an interaction described by the potential energy function $$U\left(x\right) = -Ax^4$$, where $$A > 0$$ is a constant. The particle's energy is $$E = 0$$ and its position at $$t = 0$$ is $$x_0 > 0$$.

Assume that $$v_0 < 0$$ and estimate (analytically) how long it would take for this particle to approach $$x = 0$$.

I solved the equation of motion and got $$t = \sqrt{\frac{m}{2A}}\left(\frac{1}{x} - \frac{1}{x_0}\right).$$

The time it takes for the particle to reach $$x = 0$$ is $$t = \infty$$, but it's clear to me that this isn't what's being asked for. I don't know how to approach this problem and I'd like some guidance.

## closed as off-topic by ZeroTheHero, Jon Custer, G. Smith, StephenG, John RennieJan 24 at 6:58

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "Homework-like questions should ask about a specific physics concept and show some effort to work through the problem. We want our questions to be useful to the broader community, and to future users. See our meta site for more guidance on how to edit your question to make it better" – ZeroTheHero, Jon Custer, G. Smith, StephenG, John Rennie
If this question can be reworded to fit the rules in the help center, please edit the question.

## 1 Answer

You're probably not supposed to actually solve it, just estimate it using some simple approximations. For example, you could figure out the initial velocity $$v$$ and then take $$x_0/v$$ as a rough timescale.