---
title: "A particle of mass \\(m\\) moves along the \\(x\\)-axis in a region where its motion is governed solely by a conservative force associated with the potential energy function \\(U(x)\\). The particle has total mechanical energy \\(E\\) and moves between the turning points \\(x_1\\) and \\(x_2\\), where \\(x_1  U(x)\\) for all \\(x_1 < x < x_2\\). Which of the following expressions correctly represents the time \\(\\Delta t\\) required for the particle to travel directly from \\(x_1\\) to \\(x_2\\)?"
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url: "https://nerd-notes.com/ubq/124125/"
date_modified: "2026-09-28T13:30:36+00:00"
---

# A particle of mass \(m\) moves along the \(x\)-axis in a region where its motion is governed solely by a conservative force associated with the potential energy function \(U(x)\). The particle has total mechanical energy \(E\) and moves between the turning points \(x_1\) and \(x_2\), where \(x_1  U(x)\) for all \(x_1 < x < x_2\). Which of the following expressions correctly represents the time \(\Delta t\) required for the particle to travel directly from \(x_1\) to \(x_2\)?

A particle of mass \(m\) moves along the \(x\)-axis in a region where its motion is governed solely by a conservative force associated with the potential energy function \(U(x)\). The particle has total mechanical energy \(E\) and moves between the turning points \(x_1\) and \(x_2\), where \(x_1 < x_2\) and \(E > U(x)\) for all \(x_1 < x < x_2\). Which of the following expressions correctly represents the time \(\Delta t\) required for the particle to travel directly from \(x_1\) to \(x_2\)?

![A two-dimensional plot with a horizontal axis labeled x and a vertical axis labeled U(x). A solid curve representing a potential well enters from the upper-left, curves smoothly downward to a single local minimum, and curves smoothly upward toward the upper-right. A horizontal dashed line labeled E intersects the solid curve at exactly two points: a left intersection point and a right intersection point. A vertical dashed line extends downward from the left intersection point to a tick mark on the horizontal axis labeled x_1. A second vertical dashed line extends downward from the right intersection point to a tick mark on the horizontal axis labeled x_2. In the region between x_1 and x_2, the solid curve lies entirely below the horizontal dashed line. No other curves, lines, shaded regions, gridlines, or labels appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790602235-zCbCSH.jpg)

- **A.** \(\int_{x_1}^{x_2} \sqrt{\dfrac{m}{E - U(x)}} \, dx\)
- **B.** \(\int_{x_1}^{x_2} \sqrt{\dfrac{m}{2(E - U(x))}} \, dx\)
- **C.** \(\int_{x_1}^{x_2} \sqrt{\dfrac{2m}{E - U(x)}} \, dx\)
- **D.** \(\int_{x_1}^{x_2} \sqrt{\dfrac{m}{2(U(x) - E)}} \, dx\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124125/*