Question

A rocket is fired from the earth towards the sun. At what distance from the earth’s centre is the gravitational force on the rocket zero ? Mass of the sun = 2×1030 kg, mass of the earth = 6×1024 kg. Neglect the effect of other planets etc. (orbital radius= 1.5 × 1011 m).

Answer 1

Mass of the Sun, Ms = 2 × 1030 kg
Mass of the Earth, Me = 6 × 10 24 kg
Orbital radius, r = 1.5 × 1011 m
Mass of the rocket = m

Let x be the distance from the centre of the Earth where the gravitational force acting on satellite P becomes zero.

From Newton’s law of gravitation, we can equate gravitational forces acting on satellite P under the influence of the Sun and the Earth as:

$\frac{GmM_s}{ (r-x)^2}$ = Gm $\frac{M_3}{x^2}$

$(\frac{r-x}{x})^2 =\frac{ M_s}{ M_e}$

$\frac{r-x }{x} = (\frac{2 x 10^30 }{ 60 x 10^24 })^{ \frac{1}{2} }= 577.35$

1.5 x 1011 -x = 577.35x

578.35x = 1.5 x 1011

$x = \frac{1.5 x 10^11 }{ 578.35}$ = 2.59 x 108 m