Question

A spaceship is stationed on Mars. How much energy must be expended on the spaceship to launch it out of the solar system ? Mass of the space ship = 1000 kg; mass of the sun = $2\times {10}^{30}kg$; mass of mars = $6.4\times {10}^{23}kg$; Radius of mars = 3395 km; Radius of the orbit of mars = $2.28\times {10}^{8}km$

$G=6.67\times {10}^{-11}N{m}^{2}k{g}^{-2}$.

Answer 1

Mass of the spaceship, $m_s=1000kg$

Mass of the Sun, $M=2\times {10}^{30}kg$

Mass of Mars, $m_m=6.4\times {10}^{23}kg$

Orbital radius of Mars, $R=2.28\times {10}^{11}m$

Radius of Mars, $r=3.395\times {10}^{6}m$

Universal gravitational constant, $G=6.67\times {10}^{-11}N{m}^{2}k{g}^{-2}$

The potential energy of the spaceship due to the gravitational attraction of the Sun, $U_s=-GM{m}_s/R$

Potential energy of the spaceship due to the gravitational attraction of Mars, $U_m=-G{m}_m{m}_s/r$

Since the spaceship is stationed on Mars, its velocity and hence, its kinetic energy will be zero.

Total energy of the spaceship, $E=U_m+U_s=-GM{m}_s/R-G{m}_m{m}_s/r$

The negative sign indicates that the satellite is bound to the system.

Energy required for launching the spaceship out of the solar system $=-E$

$-E=\frac{GM{m}_s}{R}+\frac{G{m}_m{m}_s}{r}$

$=G{m}_s(\frac{M}{R}+\frac{m_m}{r})$

$=6.67\times {10}^{-11}\times {10}^3\times (\frac{2\times {10}^{30}}{2.28\times {10}^{11}}+\frac{6.4\times {10}^{23}}{3.395\times {10}^6})$

$=5.91\times {10}^{11}J$