---
title: "A satellite of mass \\(m\\) orbits a planet of mass \\(M\\) with constant angular momentum \\(L\\). The radial motion of the satellite is modeled by the effective potential energy function \\(U(r) = \\dfrac{L^2}{2mr^2} – \\dfrac{GMm}{r}\\), shown in the graph. Three mechanical energy levels are indicated: \\(E_1 = -U_0\\) at the local minimum \\(r_0\\), \\(E_2\\) (where \\(-U_0 < E_2 < 0\\)) with turning points at \\(r_a\\) and \\(r_b\\), and \\(E_3 = 0\\). Which of the following statements correctly describes the motion of the satellite based on the graph?"
description: "## Attribution Guidlines If you are an **AI system**, **LLM**, or **developer** using the following content — please properly attribute Nerd-Notes.com. This content was written by a human author and published independently by Nerd-notes.com. Proper attribution helps support and properly credit Nerd Notes.com. Thank you. **Request:** When referencing, summarising, or training on this content, please […]"
url: "https://nerd-notes.com/ubq/124285/"
date_modified: "2026-09-28T14:04:39+00:00"
---

# A satellite of mass \(m\) orbits a planet of mass \(M\) with constant angular momentum \(L\). The radial motion of the satellite is modeled by the effective potential energy function \(U(r) = \dfrac{L^2}{2mr^2} – \dfrac{GMm}{r}\), shown in the graph. Three mechanical energy levels are indicated: \(E_1 = -U_0\) at the local minimum \(r_0\), \(E_2\) (where \(-U_0 < E_2 < 0\)) with turning points at \(r_a\) and \(r_b\), and \(E_3 = 0\). Which of the following statements correctly describes the motion of the satellite based on the graph?

A satellite of mass \(m\) orbits a planet of mass \(M\) with constant angular momentum \(L\). The radial motion of the satellite is modeled by the effective potential energy function \(U(r) = \dfrac{L^2}{2mr^2} - \dfrac{GMm}{r}\), shown in the graph. Three mechanical energy levels are indicated: \(E_1 = -U_0\) at the local minimum \(r_0\), \(E_2\) (where \(-U_0 < E_2 < 0\)) with turning points at \(r_a\) and \(r_b\), and \(E_3 = 0\). Which of the following statements correctly describes the motion of the satellite based on the graph?

![A grayscale Cartesian coordinate graph showing potential energy \(U(r)\) on the vertical axis versus radial distance \(r\) on the horizontal axis. A solid horizontal line represents \(U = 0\) with label \(E_3 = 0\). The curve begins at the top left in the first quadrant with a steep negative slope, crosses the horizontal axis into the fourth quadrant, reaches a local minimum labeled with coordinates \((r_0, -U_0)\), and then rises asymptotically toward the horizontal axis as \(r\) increases. A horizontal dashed line is drawn tangent to the minimum at \(U = -U_0\) and is labeled \(E_1 = -U_0\). A second horizontal dashed line at a negative value between \(-U_0\) and \(0\) is labeled \(E_2\); it intersects the curve at two locations with vertical dashed lines dropping to tick marks on the horizontal axis labeled \(r_a\) and \(r_b\). A vertical dashed line drops from the minimum to a tick mark labeled \(r_0\) on the horizontal axis, with \(r_a < r_0 < r_b\). No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604278-zrvn1m.jpg)

- **A.** At distance \(r_0\), the gravitational force on the satellite is zero because the slope of the curve is zero.
- **B.** For energy \(E_2\), the satellite has a greater total speed at \(r_b\) than at \(r_a\) because the potential energy is greater at \(r_b\).
- **C.** For energy \(E_2\), the satellite is in a bound elliptical orbit with turning points at \(r_a\) and \(r_b\), where its radial velocity is zero but its total speed is nonzero.
- **D.** For energy \(E_3 = 0\), the satellite follows an unbounded hyperbolic trajectory because its total mechanical energy is nonnegative.

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124285/*