Question

A spaceship is stationed on Mars. How much energy must be expended on the spaceship to rocket it out of the solar system? Mass of the spaceship $=1000kg$, mass of the Sun $=2\times {10}^{30}kg$, mass of the Mars $=6.4\times {10}^{23}kg$, radius of Mars $=3395km$, radius of the orbit of Mars $=2.28\times {10}^{11}m,G=6.67\times {10}^{-11}N{m}^{2}k{g}^{-2}$.

Answer 1

Potential energy of the spaceship due to the Sun.

Let $R$ be the radius of orbit of Mars and $R^{\prime }$ be the radius of the Mars. $M$ be the mass of the Sun and $M^{\prime }$ be the mass of Mars.

If $m$ is the mass of the spaceship,

Then the $PE$ of spaceship due to the Sun $=-\frac{GMm}{R}$

Potential energy of the spaceship due to the Mars.

Potential energy of spaceship due to the Mars $=-\frac{G{M}^{\prime }m}{R^{\prime }}$
The $KE$ of spaceship is zero.

Total energy of the spaceship.

Total energy of the spaceship $=-\frac{GMm}{R}-\frac{G{M}^{\prime }m}{R^{\prime }}=-Gm(\frac{M}{R}+\frac{M^{\prime }}{R^{\prime }})$


Energy required to rocket out the spaceship.

Energy required to rocket out the spaceship = −(Total energy of spaceship)

$=Gm(\frac{M}{R}+\frac{M^{\prime }}{R^{\prime }})$
$=6.67\times {10}^{-11}\times 1000\times [\frac{2\times {10}^{30}}{2.28\times {10}^{11}}+\frac{6.64\times {10}^{23}}{3395\times {10}^3}]$
$=6.67\times {10}^{-8}[\frac{20}{2.28}+\frac{6.4}{33.95}]\times {10}^{18}J=5.98\times {10}^{11}J$


$\text{Final Answer}:5.98\times {10}^{11}J$