Class 11

Physics

Modern Physics

Gravitation

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 = $2×10_{30}kg$; mass of mars = $6.4×10_{23}kg$; Radius of mars = 3395 km; Radius of the orbit of mars = $2.28×10_{8}km$

$G=6.67×10_{−11}Nm_{2}kg_{−2}$.

Mass of the spaceship, $m_{s}=1000kg$

Mass of the Sun, $M=2×10_{30}kg$

Mass of Mars, $m_{m}=6.4×10_{23}kg$

Orbital radius of Mars, $R=2.28×10_{11}m$

Radius of Mars, $r=3.395×10_{6}m$

Universal gravitational constant, $G=6.67×10_{−11}Nm_{2}kg_{−2}$

The potential energy of the spaceship due to the gravitational attraction of the Sun, $U_{s}=−GMm_{s}/R$

Potential energy of the spaceship due to the gravitational attraction of Mars, $U_{m}=−Gm_{m}m_{s}/r$

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

Total energy of the spaceship, $E=U_{m}+U_{s}=−GMm_{s}/R−Gm_{m}m_{s}/r$

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=RGMm_{s} +rGm_{m}m_{s} $

$=Gm_{s}(RM +rm_{m} )$

$=6.67×10_{−11}×10_{3}×(2.28×10_{11}2×10_{30} +3.395×10_{6}6.4×10_{23} )$

$=5.91×10_{11}J$