## Physics for Scientists and Engineers: A Strategic Approach with Modern Physics (4th Edition)

Published by Pearson

# Chapter 13 - Newton's Theory of Gravity - Exercises and Problems: 49

#### Answer

They are orbiting at an altitude of $0.414 ~R$ above the surface (where R is the planet's radius).

#### Work Step by Step

We can write an expression for the free-fall acceleration on the planet's surface. Let $M$ be the planet's mass and let $R$ be the planet's radius. $g = \frac{G~M}{R^2}$ Let $h$ be the altitude above the planet's surface. We can write an expression for the free-fall acceleration at the altitude $h$. $\frac{g}{2} = \frac{G~M}{(R+h)^2}$ $g = \frac{2~G~M}{(R+h)^2}$ We can equate the two expressions for $g$ to find $h$. $\frac{G~M}{R^2} = \frac{2~G~M}{(R+h)^2}$ $(R+h)^2 = 2~R^2$ $R+h = \sqrt{2}~R$ $h = (\sqrt{2}-1)~R$ $h = 0.414~R$ They are orbiting at an altitude of $0.414 ~R$ above the surface (where R is the planet's radius).

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