---
title: "A particle of mass \\(m\\) moves along the \\(x\\)-axis in a region where its potential energy is given by \\(U(x) = U_0\\left[\\left(\\dfrac{x}{b}\\right)^2 – \\dfrac{1}{3}\\left(\\dfrac{x}{b}\\right)^3\\right]\\), where \\(U_0\\) and \\(b\\) are positive constants.  Particle 1 is released from rest at \\(x = -b\\), and an identical Particle 2 is released from rest at \\(x = +b\\). What is the ratio of the maximum speed of Particle 1 to the maximum speed of Particle 2, \\(\\dfrac{v_{1,\\text{max}}}{v_{2,\\text{max}}}\\)?"
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url: "https://nerd-notes.com/ubq/124090/"
date_modified: "2026-09-28T13:30:23+00:00"
---

# A particle of mass \(m\) moves along the \(x\)-axis in a region where its potential energy is given by \(U(x) = U_0\left[\left(\dfrac{x}{b}\right)^2 – \dfrac{1}{3}\left(\dfrac{x}{b}\right)^3\right]\), where \(U_0\) and \(b\) are positive constants.

Particle 1 is released from rest at \(x = -b\), and an identical Particle 2 is released from rest at \(x = +b\). What is the ratio of the maximum speed of Particle 1 to the maximum speed of Particle 2, \(\dfrac{v_{1,\text{max}}}{v_{2,\text{max}}}\)?

A particle of mass \(m\) moves along the \(x\)-axis in a region where its potential energy is given by \(U(x) = U_0\left[\left(\dfrac{x}{b}\right)^2 - \dfrac{1}{3}\left(\dfrac{x}{b}\right)^3\right]\), where \(U_0\) and \(b\) are positive constants.

Particle 1 is released from rest at \(x = -b\), and an identical Particle 2 is released from rest at \(x = +b\). What is the ratio of the maximum speed of Particle 1 to the maximum speed of Particle 2, \(\dfrac{v_{1,\text{max}}}{v_{2,\text{max}}}\)?

![A Cartesian coordinate graph with horizontal axis labeled x and vertical axis labeled U(x). The horizontal axis has tick marks labeled -b, 0, b, and 2b. A single solid curve represents the potential energy function. The curve decreases from the second quadrant to a minimum at the origin (0, 0), rises through the first quadrant passing through x = b, reaches a local maximum at x = 2b, and then turns downward. Two release states are marked on the curve with solid circular dots: one dot is located at x = -b at a vertical height corresponding to U = \dfrac{4}{3}U_0, and a second dot is located at x = +b at a lower vertical height corresponding to U = \dfrac{2}{3}U_0. Bare axes with no gridlines. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790602222-ifQlcf.jpg)

- **A.** \(\sqrt{2}\)
- **B.** \(\sqrt{3}\)
- **C.** \(2\)
- **D.** \(3\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124090/*