Question

1. if the distance between a spaceship and the center of the earth is increased from one earth radius to 4 earth radii, the gravitational force acting on the spaceship becomes approximately

a. $\frac{1}{16}$as great.
b. $\frac{1}{4}$as great.
c. 16times greater.
d. 4

Explanation:

Step1: Recall gravitational - force formula

The gravitational force formula is $F = \frac{GMm}{r^{2}}$, where $G$ is the gravitational constant, $M$ is the mass of the Earth, $m$ is the mass of the spaceship, and $r$ is the distance between the spaceship and the center of the Earth.

Step2: Consider initial and final distances

Let the initial distance $r_1 = R_E$ (Earth - radius) and the final distance $r_2=4R_E$. The initial gravitational force $F_1=\frac{GMm}{R_E^{2}}$, and the final gravitational force $F_2=\frac{GMm}{(4R_E)^{2}}=\frac{GMm}{16R_E^{2}}$.

Step3: Find the ratio of forces

Calculate the ratio $\frac{F_2}{F_1}=\frac{\frac{GMm}{16R_E^{2}}}{\frac{GMm}{R_E^{2}}}=\frac{1}{16}$.

Answer:

A. $\frac{1}{16}$ as great.