# 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

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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.