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title: "A research probe of mass \\(m\\) is launched vertically upward from the surface of a planet. The magnitude of the gravitational acceleration experienced by the probe varies with altitude \\(z\\) according to \\(g(z) = \\dfrac{g_0}{\\left(1 + \\dfrac{z}{R}\\right)^2}\\), where \\(g_0\\) is the surface gravitational acceleration at \\(z = 0\\) and \\(R\\) is a constant. Assuming atmospheric drag is negligible, what is the work done by the upward external force required to lift the probe at constant speed from \\(z = 0\\) to an altitude \\(z = H\\)?"
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url: "https://nerd-notes.com/ubq/117639/"
date_modified: "2026-08-04T07:49:26+00:00"
---

# A research probe of mass \(m\) is launched vertically upward from the surface of a planet. The magnitude of the gravitational acceleration experienced by the probe varies with altitude \(z\) according to \(g(z) = \dfrac{g_0}{\left(1 + \dfrac{z}{R}\right)^2}\), where \(g_0\) is the surface gravitational acceleration at \(z = 0\) and \(R\) is a constant. Assuming atmospheric drag is negligible, what is the work done by the upward external force required to lift the probe at constant speed from \(z = 0\) to an altitude \(z = H\)?

A research probe of mass \(m\) is launched vertically upward from the surface of a planet. The magnitude of the gravitational acceleration experienced by the probe varies with altitude \(z\) according to \(g(z) = \dfrac{g_0}{\left(1 + \dfrac{z}{R}\right)^2}\), where \(g_0\) is the surface gravitational acceleration at \(z = 0\) and \(R\) is a constant. Assuming atmospheric drag is negligible, what is the work done by the upward external force required to lift the probe at constant speed from \(z = 0\) to an altitude \(z = H\)?

![A vertical z-axis pointing upward with origin z = 0 at a horizontal shaded surface. A small square block labeled m is positioned above the surface at altitude z = H. An upward arrow labeled F_ext and a downward arrow labeled F_g are attached to mass m. Dotted horizontal lines mark z = 0 and z = H along the z-axis. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785829766-zmVg2j.jpg)

- **A.** \(\dfrac{m g_0 H}{\left(1 + \dfrac{H}{R}\right)^2}\)
- **B.** \(m g_0 H \left(1 + \dfrac{H}{R}\right)\)
- **C.** \(\dfrac{m g_0 H}{1 + \dfrac{H}{R}}\)
- **D.** \(\dfrac{m g_0 H^2}{R \left(1 + \dfrac{H}{R}\right)}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117639/*