---
title: "A particle of mass \\(m = 0.20 \\text{ kg}\\) moves along the \\(x\\)-axis subject only to a conservative force with the potential energy function \\(U(x)\\) shown in the graph. The horizontal dashed line represents a constant total mechanical energy of \\(E = -2.0 \\text{ J}\\). Which of the following statements correctly characterizes the motion and dynamical properties of the particle?"
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url: "https://nerd-notes.com/ubq/124066/"
date_modified: "2026-09-28T13:30:19+00:00"
---

# A particle of mass \(m = 0.20 \text{ kg}\) moves along the \(x\)-axis subject only to a conservative force with the potential energy function \(U(x)\) shown in the graph. The horizontal dashed line represents a constant total mechanical energy of \(E = -2.0 \text{ J}\). Which of the following statements correctly characterizes the motion and dynamical properties of the particle?

A particle of mass \(m = 0.20 \text{ kg}\) moves along the \(x\)-axis subject only to a conservative force with the potential energy function \(U(x)\) shown in the graph. The horizontal dashed line represents a constant total mechanical energy of \(E = -2.0 \text{ J}\). Which of the following statements correctly characterizes the motion and dynamical properties of the particle?

![A Cartesian graph with a horizontal axis labeled x (m) extending from 0 to 8 with integer tick marks at 1, 2, 3, 4, 5, 6, 7, 8, and a vertical axis labeled U (J) extending from -6 to 4 with tick marks at -6, -4, -2, 0, 2, 4. Light gridlines align with each tick mark. A horizontal dashed line is drawn across the graph at U = -2, labeled E = -2.0 J at the right edge. A single smooth, solid curve representing U(x) enters from the top near x = 0.5, decreases steeply, crosses U = 0 at x = 1.0, intersects the dashed line at x = 1.5, reaches a local minimum at (2.0, -6), rises and intersects the dashed line at x = 3.5, reaches a local maximum at (5.0, -1), decreases and intersects the dashed line at x = 6.5, and then flattens asymptotically toward U = -4 as x approaches 8. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790602219-npuloz.jpg)

- **A.** The particle can oscillate continuously between turning points at \(x = 1.5 \text{ m}\) and \(x = 6.5 \text{ m}\), experiencing a net force directed in the \(+x\)-direction throughout the interval \(2.0 \text{ m} < x < 5.0 \text{ m}\).
- **B.** The particle experiences zero net force at \(x = 1.0 \text{ m}\) where the potential energy is zero, and reaches its maximum speed at \(x = 5.0 \text{ m}\) where the potential energy barrier reaches a local maximum.
- **C.** If the particle is located in the interval \(1.5 \text{ m} \le x \le 3.5 \text{ m}\), its motion is bounded between two turning points, and its maximum kinetic energy in this interval is strictly greater than the kinetic energy it could have anywhere in the region \(x \ge 6.5 \text{ m}\).
- **D.** If the particle is located in the region \(x \ge 6.5 \text{ m}\), its motion is bounded because \(U(x)\) approaches a horizontal asymptote, and its kinetic energy increases without bound as \(x \to \infty\).

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124066/*