Question

An astronaut is in equilibrium when he is positioned 140 km from the center of asteroid X and 581 km from the center of asteroid Y, along the straight line joining the centers of the asteroids. What is the ratio of the masses X/Y of the asteroids? Show all work and circle answe

Answer 1

Answer:

0.05806

Explanation:

$m_x$ = Mass of asteroid x

$m_y$ = Mass of asteroid y

$r_x$ = Distance from asteroid x = 140 km

$r_y$ = Distance from asteroid y = 581 km

m = Mass of asteroid

Force of gravity between asteroid x and the astronaut

$F_1=\frac{Gm_xm}{r_x^2}\\\Rightarrow F_1=\frac{Gm_xm}{140^2}$

Force of gravity between asteroid x and the astronaut

$F_2=\frac{Gm_ym}{r_y^2}\\\Rightarrow F_2=\frac{Gm_ym}{581^2}$

Here these two forces are equal as they are in equilibrium

$\frac{Gm_xm}{140^2}=\frac{Gm_ym}{581^2}\\\Rightarrow \frac{m_x}{140^2}=\frac{m_y}{581^2}\\\Rightarrow \frac{m_x}{m_y}=\frac{140^2}{581^2}\\\Rightarrow \frac{m_x}{m_y}=0.05806$

The ratio of the masses of the asteroid is 0.05806