Question

What is the distance of the spaceship from the earth

Answer 1

Gravitational force on spaceship is given by equation

`F = (Gm_1m_2)/(r^2)`

here
`m_1, m_2` = masses of two objects

r = distance between their centers

now it is given that gravitation force due to earth is two times the force of gravitation due to moon

`F_(earth) =2 F_(moon)`

`(GM_e m)/(r^2) =2 (GM_m m)/((d-r)^2)`

`(M_e)/(r^2) =2 (M_m )/((d-r)^2)`

`M_e (d -r)^2 =2 M_m r^2`

`(\sqrt{2(M_m)/(M_e)} + 1)r = d`

given that

`M_m = 7.35 * 10^(22) kg`

`M_e = 5.98 * 10^(24) kg`

d = 384000 km

now from above equation

`(\sqrt{2(7.35* 10^(22))/(5.98*10^(24))} + 1)r = 384000 km`

r = 269875.5 km

so the distance from earth will be 269875.5 km