Two 65 kg astronauts leave earth in a spacecraft, sitting 1.0 m apart.

Question

How far are they from the center of the earth when the gravitational force between them is as strong as the gravitational force of the earth on one of the astronauts?

Answer 1

 

m = mass of astronauts (the same for both) = 65 kg

r = distance between the astronauts = 1.0 m

M = mass of earth = 5.98 x 10²⁴ kg

G = universal constant of gravitation (it will drop out of the solution)

R = distance from astronaut to center of earth (what we want to solve for)

 

Gm²/r² = gravitational force between astronauts

 

GMm/R² = gravitational force between an astronaut and the earth

 

Gm²/r² = GMm/R²

 

m/r² = M/R²

 

R² = (M/m)r²

 

R = [√(M/m)] r = √[(5.98 x 10²⁴)/(65)] (1.0 m) ≅ 3.03 x 10¹¹ m