---
title: "A satellite of mass \\(m\\) orbits a planet of mass \\(M\\) in an initial circular orbit of radius \\(R\\) with orbital speed \\(v_0\\). The satellite transitions to a larger coplanar circular orbit of radius \\(4R\\) via a Hohmann transfer ellipse. At point \\(P\\) on the initial orbit, an instantaneous tangential forward thrust injects the satellite into the elliptical transfer orbit, which has periapsis distance \\(R\\) at point \\(P\\) and apoapsis distance \\(4R\\) at point \\(A\\). In terms of \\(m\\), \\(R\\), and \\(v_0\\), what are the magnitude of the satellite’s angular momentum \\(L_{\\text{trans}}\\) about the planet’s center during the transfer orbit and the work \\(W_1\\) done on the satellite by the engine thrust during this initial burn?"
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url: "https://nerd-notes.com/ubq/120939/"
date_modified: "2026-08-23T04:44:28+00:00"
---

# A satellite of mass \(m\) orbits a planet of mass \(M\) in an initial circular orbit of radius \(R\) with orbital speed \(v_0\). The satellite transitions to a larger coplanar circular orbit of radius \(4R\) via a Hohmann transfer ellipse. At point \(P\) on the initial orbit, an instantaneous tangential forward thrust injects the satellite into the elliptical transfer orbit, which has periapsis distance \(R\) at point \(P\) and apoapsis distance \(4R\) at point \(A\). In terms of \(m\), \(R\), and \(v_0\), what are the magnitude of the satellite’s angular momentum \(L_{\text{trans}}\) about the planet’s center during the transfer orbit and the work \(W_1\) done on the satellite by the engine thrust during this initial burn?

A satellite of mass \(m\) orbits a planet of mass \(M\) in an initial circular orbit of radius \(R\) with orbital speed \(v_0\). The satellite transitions to a larger coplanar circular orbit of radius \(4R\) via a Hohmann transfer ellipse. At point \(P\) on the initial orbit, an instantaneous tangential forward thrust injects the satellite into the elliptical transfer orbit, which has periapsis distance \(R\) at point \(P\) and apoapsis distance \(4R\) at point \(A\). In terms of \(m\), \(R\), and \(v_0\), what are the magnitude of the satellite's angular momentum \(L_{\text{trans}}\) about the planet's center during the transfer orbit and the work \(W_1\) done on the satellite by the engine thrust during this initial burn?

![A schematic diagram showing orbital trajectories around a central planet. At the center of the frame, a solid filled circle represents a planet of mass \(M\). An inner dashed circle of radius \(R\) represents the initial orbit, passing through point \(P\) located directly to the right of the planet. An outer dashed circle of radius \(4R\) represents the final circular orbit, passing through point \(A\) located directly to the left of the planet. A solid ellipse with one focus at the planet represents the transfer orbit; its periapsis touches the inner circle at point \(P\) and its apoapsis touches the outer circle at point \(A\). A straight horizontal dimension line extends from the planet's center to point \(P\), labeled \(R\). Another straight horizontal dimension line extends from the planet's center to point \(A\), labeled \(4R\). An arrow at point \(P\) points tangentially along the transfer ellipse in the counterclockwise direction. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787460268-VHdA75.jpg)

- **A.** \(L_{\text{trans}} = \sqrt{\dfrac{5}{3}} m R v_0\) and \(W_1 = \dfrac{1}{3} m v_0^2\)
- **B.** \(L_{\text{trans}} = \sqrt{\dfrac{8}{5}} m R v_0\) and \(W_1 = \dfrac{3}{10} m v_0^2\)
- **C.** \(L_{\text{trans}} = \sqrt{\dfrac{8}{5}} m R v_0\) and \(W_1 = \dfrac{3}{5} m v_0^2\)
- **D.** \(L_{\text{trans}} = \dfrac{3}{2} m R v_0\) and \(W_1 = \dfrac{3}{10} m v_0^2\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/120939/*