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 = 2×1030 kg; mass of mars = 6.4×1023 kg; radius of mars = 3395 km; radius of the orbit of mars = 2.28 ×108 km; G = 6.67×10-11 N m2kg–2.
Mass of the spaceship, ms=1000 kg Mass of the Sun, M=2×1030 kg Mass of Mars, mm=6.4×1023 kg Orbital radius of Mars, R=2.28×1011 m Radius of Mars, r=3.395×106 m Universal gravitational constant, G=6.67×10−11 Nm2kg−2 The potential energy of the spaceship due to the gravitational attraction of the Sun, Us=−GMms/R Potential energy of the spaceship due to the gravitational attraction of Mars, Um=−Gmmms/r Since the spaceship is stationed on Mars, its velocity and hence, its kinetic energy will be zero. Total energy of the spaceship, E=Um+Us=−GMms/R−Gmmms/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=RGMms+rGmmms =Gms(RM+rmm) =6.67×10−11×103×(2.28×10112×1030+3.395×1066.4×1023) =5.91×1011 J
