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).

Solution

Given,
Mass of the sun, M = 2× 10³⁰ kg
Mass of the earth, m = 6.0 × 10²⁴ kg
Distance from the sun and the earth is,
   $straight r equals 1.5 cross times 10 to the power of 11 straight m$

Let the force on the rocket be zero at a distance x from the earth.
∴ $space space fraction numerator GMm subscript 0 over denominator left parenthesis straight r minus straight x right parenthesis squared end fraction equals Gmm subscript 0 over straight x squared$
where, $straight m subscript 0$ is a mass of the rocket.
Therefore,
   $fraction numerator straight M over denominator left parenthesis straight r minus straight x right parenthesis squared end fraction equals straight m over straight x squared$

$rightwards double arrow$
$rightwards double arrow$    $straight r over straight x equals 1 plus square root of straight M over straight m end root equals 1 plus square root of fraction numerator 2 cross times 10 to the power of 30 over denominator 60 cross times 10 to the power of 24 end fraction end root$
= 578.35
Therefore,
Gravitational force is zero on the rocket at a distance of,
$space space straight x equals fraction numerator straight r over denominator 587.35 end fraction equals fraction numerator 1.5 cross times 10 to the power of 11 over denominator 587.35 end fraction straight m$
$space space equals 2.6 cross times 10 to the power of 8 straight m$ , from the Earth's surface.