Question

A rocket is fired from the earth towards the moon. At what distance from the moon is the gravitational force on the rocket zero? Mass of earth = 6 x 10²⁴ kg and Mass of moon = 7.4 x 10²² kg and orbital radius of the moon= 3.8 x 10⁸ m. Neglect the effect of the sun and the other planets.

Answer 1

Let the distance between the Earth and Moon be r.

Now,

There are two opposite forces acting on the rocket of mass m,

gravitational force of the earth = GM_em / x²

where M_e is the mass of the earth and x is the distance between the rocket and earth, and

gravitational force of the moon = GM_m / (r-x)²

now,

for the rocket to experience no force

GM_em / x² = GM_m / (r-x)²

or

(r-x)² = (M_m /M_e)x²

now,

r = 3.8 x 10⁸ m (not radius of moon)

M_m = 7.4 x 10²² kg

M_e= 7.4 x 10²² kg

so,

(r-x)² = (7.4 x 10²² / 7.4 x 10²² ).x²

or solving further

r - x = 0.11x

thus,

x = r / 1.11 = 3.8 x 10⁸ / 1.11

so,

x = 3.468 x 10⁸ m

thus, the distance from the moon will be

r-x = 3.8 x 10⁸ m - 3.468 x 10⁸ m

thus,

r-x = 3.82 x 10⁷ m