---
title: "A particle of mass \\(m\\) moves along the positive radial direction under the influence of a conservative central force. The potential energy of the particle as a function of separation distance \\(r\\) is given by  \\[ U(r) = 4\\varepsilon \\left[ \\left(\\dfrac{\\sigma}{r}\\right)^{12} – \\left(\\dfrac{\\sigma}{r}\\right)^6 \\right] \\]  where \\(\\varepsilon\\) and \\(\\sigma\\) are positive constants. The particle is initially located at its stable equilibrium separation \\(r_{\\text{eq}}\\). In terms of \\(\\varepsilon\\), \\(m\\), and given constants, what is the minimum initial speed \\(v_{\\text{esc}}\\) that must be given to the particle so that it can escape to an infinitely large separation (\\(r \\to \\infty\\))?"
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url: "https://nerd-notes.com/ubq/120710/"
date_modified: "2026-08-23T04:42:48+00:00"
---

# A particle of mass \(m\) moves along the positive radial direction under the influence of a conservative central force. The potential energy of the particle as a function of separation distance \(r\) is given by

\[ U(r) = 4\varepsilon \left[ \left(\dfrac{\sigma}{r}\right)^{12} – \left(\dfrac{\sigma}{r}\right)^6 \right] \]

where \(\varepsilon\) and \(\sigma\) are positive constants. The particle is initially located at its stable equilibrium separation \(r_{\text{eq}}\). In terms of \(\varepsilon\), \(m\), and given constants, what is the minimum initial speed \(v_{\text{esc}}\) that must be given to the particle so that it can escape to an infinitely large separation (\(r \to \infty\))?

A particle of mass \(m\) moves along the positive radial direction under the influence of a conservative central force. The potential energy of the particle as a function of separation distance \(r\) is given by

\[ U(r) = 4\varepsilon \left[ \left(\dfrac{\sigma}{r}\right)^{12} - \left(\dfrac{\sigma}{r}\right)^6 \right] \]

where \(\varepsilon\) and \(\sigma\) are positive constants. The particle is initially located at its stable equilibrium separation \(r_{\text{eq}}\). In terms of \(\varepsilon\), \(m\), and given constants, what is the minimum initial speed \(v_{\text{esc}}\) that must be given to the particle so that it can escape to an infinitely large separation (\(r \to \infty\))?

![A 2D line graph with a horizontal axis labeled r and a vertical axis labeled U(r). The horizontal axis has tick mark \sigma where the curve crosses zero. The vertical axis has tick mark 0 where the axes cross. A solid curve starts at high positive U near the vertical axis, descends steeply toward the right, crosses the horizontal axis at r = \sigma, reaches a smooth local minimum in the fourth quadrant, and then rises smoothly from below to asymptotically approach U = 0 as r increases to the right. A vertical dashed line extends upward from the minimum to the horizontal axis at label r_{\text{eq}}. A horizontal dashed line extends leftward from the minimum to the vertical axis at label U_{\text{min}}. No gridlines appear. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787460168-Llma1b.jpg)

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

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/120710/*