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

At a distance of $3.0\times 10^{11}~m$, the gravitational force between the astronauts is as strong as the gravitational force of the earth on one of the astronauts.
Let $M_a$ be the mass of each astronaut. Let $M_e$ be the earth's mass. Let $R$ be the astronauts' distance from the center of the earth. To find the required distance $R$, we can equate the gravitational force $F_a$ between the astronauts to the gravitational force $F_e$ of the earth on one of the astronauts. $F_a = F_e$ $\frac{G~M_a~M_a}{(1.0~m)^2} = \frac{G~M_e~M_a}{R^2}$ $R^2 = \frac{(1.0~m)^2~M_e}{M_a}$ $R = \sqrt{\frac{(1.0~m)^2~M_e}{M_a}}$ $R = \sqrt{\frac{(1.0~m)^2~(5.98\times 10^{24}~kg)}{65~kg}}$ $R = 3.0\times 10^{11}~m$ At a distance of $3.0\times 10^{11}~m$, the gravitational force between the astronauts is as strong as the gravitational force of the earth on one of the astronauts.