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title: "Two satellites, \\(A\\) and \\(B\\), orbit a planet of mass \\(M\\) where \\(M \\gg m\\). Satellite \\(A\\) has mass \\(m\\) and moves in a circular orbit of radius \\(R_0\\). Satellite \\(B\\) has mass \\(2m\\) and moves in an elliptical orbit with semi-major axis \\(a = R_0\\), periapsis distance \\(r_p = \\dfrac{1}{2}R_0\\), and apoapsis distance \\(r_a = \\dfrac{3}{2}R_0\\). Which of the following correctly pairs the relationship between the total mechanical energies \\(E_A\\) and \\(E_B\\) of the two satellite-planet systems with the ratio of their orbital angular momentum magnitudes \\(\\dfrac{L_B}{L_A}\\)?"
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url: "https://nerd-notes.com/ubq/120920/"
date_modified: "2026-08-23T04:44:19+00:00"
---

# Two satellites, \(A\) and \(B\), orbit a planet of mass \(M\) where \(M \gg m\). Satellite \(A\) has mass \(m\) and moves in a circular orbit of radius \(R_0\). Satellite \(B\) has mass \(2m\) and moves in an elliptical orbit with semi-major axis \(a = R_0\), periapsis distance \(r_p = \dfrac{1}{2}R_0\), and apoapsis distance \(r_a = \dfrac{3}{2}R_0\). Which of the following correctly pairs the relationship between the total mechanical energies \(E_A\) and \(E_B\) of the two satellite-planet systems with the ratio of their orbital angular momentum magnitudes \(\dfrac{L_B}{L_A}\)?

Two satellites, \(A\) and \(B\), orbit a planet of mass \(M\) where \(M \gg m\). Satellite \(A\) has mass \(m\) and moves in a circular orbit of radius \(R_0\). Satellite \(B\) has mass \(2m\) and moves in an elliptical orbit with semi-major axis \(a = R_0\), periapsis distance \(r_p = \dfrac{1}{2}R_0\), and apoapsis distance \(r_a = \dfrac{3}{2}R_0\). Which of the following correctly pairs the relationship between the total mechanical energies \(E_A\) and \(E_B\) of the two satellite-planet systems with the ratio of their orbital angular momentum magnitudes \(\dfrac{L_B}{L_A}\)?

![A schematic diagram showing a large central circular disk labeled M at the center. A solid circular path of radius R_0 is drawn centered on M, with a small labeled dot A on the circular path. An aligned dashed elliptical path surrounds M such that M is located at the left focal point of the ellipse. The semi-major axis of the dashed ellipse is labeled a = R_0. A horizontal reference line passes from the left focus through the periapsis labeled r_p = \dfrac{1}{2}R_0 to the left and apoapsis labeled r_a = \dfrac{3}{2}R_0 to the right. A small labeled dot B is located at the periapsis position on the dashed elliptical path. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787460259-JVJFvG.jpg)

- **A.** Total energy: \(E_B = E_A\) ; Angular momentum ratio: \(\dfrac{L_B}{L_A} = 2\)
- **B.** Total energy: \(E_B = E_A\) ; Angular momentum ratio: \(\dfrac{L_B}{L_A} = \sqrt{3}\)
- **C.** Total energy: \(E_B = 2E_A\) ; Angular momentum ratio: \(\dfrac{L_B}{L_A} = 2\)
- **D.** Total energy: \(E_B = 2E_A\) ; Angular momentum ratio: \(\dfrac{L_B}{L_A} = \sqrt{3}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/120920/*