The radius of the earth is R.At what distance above the earth's surface will the acceleration of gravity be 4.9 m/s²? A) 0.41R B) 0.50R C) 1.0R D) 1.4R E) 2.0R

Question

Quizplus · Accepted Answer

The acceleration due to gravity at a distance $r$ from the center of the Earth (where $r > R$, $R$ being the Earth's radius) is given by $g' = \frac{G M}{r^2}$, where $G$ is the gravitational constant and $M$ is the mass of the Earth. At the surface of the Earth, the acceleration due to gravity is $g = \frac{G M}{R^2}$. If the acceleration of gravity at some height $h$ above the Earth's surface is $4.9 \, \text{m/s}^2$, which is half of the standard $9.8 \, \text{m/s}^2$, we can set up the equation $\frac{G M}{(R+h)^2} = \frac{1}{2} \frac{G M}{R^2}$. Simplifying, we get $(R+h)^2 = 2R^2$, which leads to $R+h = R\sqrt{2}$, and thus $h = R(\sqrt{2}-1)$. This is approximately $0.41R$, hence the correct answer is A.

The Radius of the Earth Is R