---
title: "A space probe of mass \\(m\\) is launched vertically upward from the surface of a spherical planet of mass \\(M\\) and radius \\(R\\) that has no atmosphere.  In Scenario 1, the probe is launched with an initial kinetic energy \\(K_1\\) such that it reaches a maximum altitude of \\(3R\\) above the planet’s surface before momentarily coming to rest.  In Scenario 2, an identical probe is launched vertically upward from the surface with the minimum initial kinetic energy \\(K_2\\) required to completely escape the gravitational pull of the planet.  What is the ratio \\(K_1 / K_2\\)?"
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url: "https://nerd-notes.com/ubq/124162/"
date_modified: "2026-09-28T13:30:51+00:00"
---

# A space probe of mass \(m\) is launched vertically upward from the surface of a spherical planet of mass \(M\) and radius \(R\) that has no atmosphere.

In Scenario 1, the probe is launched with an initial kinetic energy \(K_1\) such that it reaches a maximum altitude of \(3R\) above the planet’s surface before momentarily coming to rest.

In Scenario 2, an identical probe is launched vertically upward from the surface with the minimum initial kinetic energy \(K_2\) required to completely escape the gravitational pull of the planet.

What is the ratio \(K_1 / K_2\)?

A space probe of mass \(m\) is launched vertically upward from the surface of a spherical planet of mass \(M\) and radius \(R\) that has no atmosphere.

In Scenario 1, the probe is launched with an initial kinetic energy \(K_1\) such that it reaches a maximum altitude of \(3R\) above the planet's surface before momentarily coming to rest.

In Scenario 2, an identical probe is launched vertically upward from the surface with the minimum initial kinetic energy \(K_2\) required to completely escape the gravitational pull of the planet.

What is the ratio \(K_1 / K_2\)?

- **A.** \(\dfrac{1}{4}\)
- **B.** \(\dfrac{2}{3}\)
- **C.** \(\dfrac{3}{4}\)
- **D.** \(\dfrac{15}{16}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124162/*