---
title: "A particle of mass \\(m\\) moves along the \\(x\\)-axis in a region where its potential energy is given by the function \\(U(x) = ax^4 – bx^2\\), where \\(a\\) and \\(b\\) are positive constants. The particle is initially located at one of its stable equilibrium positions. What is the minimum initial speed \\(v_0\\) that must be imparted to the particle so that it can reach the origin \\(x = 0\\)?"
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url: "https://nerd-notes.com/ubq/117598/"
date_modified: "2026-08-04T07:49:16+00:00"
---

# A particle of mass \(m\) moves along the \(x\)-axis in a region where its potential energy is given by the function \(U(x) = ax^4 – bx^2\), where \(a\) and \(b\) are positive constants. The particle is initially located at one of its stable equilibrium positions. What is the minimum initial speed \(v_0\) that must be imparted to the particle so that it can reach the origin \(x = 0\)?

A particle of mass \(m\) moves along the \(x\)-axis in a region where its potential energy is given by the function \(U(x) = ax^4 - bx^2\), where \(a\) and \(b\) are positive constants. The particle is initially located at one of its stable equilibrium positions. What is the minimum initial speed \(v_0\) that must be imparted to the particle so that it can reach the origin \(x = 0\)?

![A graph of potential energy U versus position x showing a symmetrical double-well potential curve. The vertical axis is labeled U and the horizontal axis is labeled x. The curve starts high in the upper-left quadrant, decreases to a local minimum at a negative position labeled -x_min, rises to a local maximum at the origin (0,0), drops to an identical local minimum at a positive position labeled x_min, and then rises steeply into the upper-right quadrant. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/potential-energy-curve-1785829756-STEbPE.jpg)

- **A.** \(\dfrac{b}{2\sqrt{am}}\)
- **B.** \(\dfrac{b}{\sqrt{am}}\)
- **C.** \(\dfrac{b}{\sqrt{2am}}\)
- **D.** \(\dfrac{b^2}{2am}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117598/*