Question

A rocket is fired from the Earth towards the Sun. At what distance from the Earth's center, the gravitational force on the rocket is zero? Mass of the Sun $=2\times {10}^{30}kg$ and mass of the Earth $=6\times {10}^{24}kg$.

• A. $2.6\times {10}^{8}m$
• B. $3.2\times {10}^{8}m$
• C. $3.9\times {10}^{9}m$
• D. $2.3\times {10}^{9}m$

Answer 1

The correct option is A $2.6\times {10}^{8}m$

Let $P$ be a point at a distance $r$ from the Earth's center, where gravitational force due to Sun and Earth are equal and opposite and hence, gravitational force on the rocket is zero.
$\frac{G{M}_{s}m}{{(x-r)}^2}=\frac{G{M}_{e}m}{r^2}\Rightarrow \frac{G{M}_s}{{(x-r)}^2}=\frac{G{M}_e}{r^2}$
$\frac{2\times {10}^{30}}{{(x-r)}^2}=\frac{6\times {10}^{24}}{r^2}$
$\frac{{(x-r)}^2}{r^2}=\frac{2\times {10}^{30}}{6\times {10}^{24}}=\frac{1}{3}\times {10}^6$
$\Rightarrow \frac{x-r}{r}=\frac{{10}^3}{\sqrt{3}}\Rightarrow r=\left(\frac{3}{1735}\right)x$
$\Rightarrow r=\frac{3}{1735}\times 1.5\times {10}^{11}=\frac{3\times 15\times 100\times {10}^8}{1735}$
$r=2.594\times {10}^{8}m$
$r=2.6\times {10}^{8}m$ from Earth.