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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 space ship $=1000 \mathrm{~kg}$, Mass of the sun $=2 \times 10^{30} \mathrm{~kg}$, Mass of the Mars $=6.4 \times$ $10^{23} \mathrm{~kg}$, Radius of Mars $=3395 \mathrm{~km}$, Radius of the orbit of Mars $=2.28 \times 10^{11} \mathrm{~m} \mathrm{G}=6.67 \times 10^{-11} \mathrm{Nm}^2 / \mathrm{kg}^2$.
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Let $R^{\prime}=$ Radius of the Mars, $R=$ Radius of the orbit of Mars, $M=$ Mass of the sun
$M^{\prime}=$ Mass of the Mars, $m=$ Mass of the space-ship.
$\therefore \quad$ P.E. of space-ship due to gravitational attraction of the Sun $=\frac{-G M m}{R}$
P.E. of space-ship due to gravitational attraction of Mars $=\frac{-G M ' m}{R^{\prime}}$
$\because \quad$ The K.E. of space-ship is zero.
$\therefore$ Total energy of the ship $=\frac{-G M m}{R}-\frac{G M^{\prime} m}{R^{\prime}}=-G m\left(\frac{M}{R}+\frac{M^{\prime}}{R^{\prime}}\right)$
Energy required to rocket out the space ship from the solar system $=-$ (total energy)
$$
\begin{aligned}
&=-\left[-G m\left(\frac{M}{R}+\frac{M^{\prime}}{R^{\prime}}\right)\right]=G m\left(\frac{M}{R}+\frac{M^{\prime}}{R^{\prime}}\right) \\
&=6.67 \times 10^{-11} \times 1000\left[\frac{2 \times 10^{30}}{2.28 \times 10^{11}}+\frac{6.4 \times 10^{23}}{3395 \times 10^3}\right] \\
&=6.67 \times 10^{-8}\left[\frac{20}{2.28}+\frac{6.4}{33.95}\right] \times 10^{18} \\
&=5.98 \times 10^{11} \mathrm{~J} .
\end{aligned}
$$
$M^{\prime}=$ Mass of the Mars, $m=$ Mass of the space-ship.
$\therefore \quad$ P.E. of space-ship due to gravitational attraction of the Sun $=\frac{-G M m}{R}$
P.E. of space-ship due to gravitational attraction of Mars $=\frac{-G M ' m}{R^{\prime}}$
$\because \quad$ The K.E. of space-ship is zero.
$\therefore$ Total energy of the ship $=\frac{-G M m}{R}-\frac{G M^{\prime} m}{R^{\prime}}=-G m\left(\frac{M}{R}+\frac{M^{\prime}}{R^{\prime}}\right)$
Energy required to rocket out the space ship from the solar system $=-$ (total energy)
$$
\begin{aligned}
&=-\left[-G m\left(\frac{M}{R}+\frac{M^{\prime}}{R^{\prime}}\right)\right]=G m\left(\frac{M}{R}+\frac{M^{\prime}}{R^{\prime}}\right) \\
&=6.67 \times 10^{-11} \times 1000\left[\frac{2 \times 10^{30}}{2.28 \times 10^{11}}+\frac{6.4 \times 10^{23}}{3395 \times 10^3}\right] \\
&=6.67 \times 10^{-8}\left[\frac{20}{2.28}+\frac{6.4}{33.95}\right] \times 10^{18} \\
&=5.98 \times 10^{11} \mathrm{~J} .
\end{aligned}
$$
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