A space-ship 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 = 1000 kg, Mass of the sun = 2 xx 10^30kg. Mass of Mars =6.4xx 10^23 kg, Radius of Mars = 3395 km. Radius of the orbit of Mars = 2.28 xx 10^11m, G=6.67 xx 10^(-11) Nm^2 kg^(-2)

A space-ship is stationed on Mars. How much energy must be expended on

1.	A space-ship 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 = 1000 kg, Mass of the sun = 2 xx 10^30kg. Mass of Mars =6.4xx 10^23 kg, Radius of Mars = 3395 km. Radius of the orbit of Mars = 2.28 xx 10^11m, G=6.67 xx 10^(-11) Nm^2 kg^(-2)
Answer» Solution : Let R.be the radius of orbit of Mars and R. be the radius of Mars. M be the MASS of the Sun and M. be the mass of Mars. If m is the mass of the space-ship, then Potential energy of space-ship due to gravitational attraction of the Sun = -GMm/R Potential energy of space-ship due to gravitational attraction of Mars =-GM.m/R. Since the K.E. of space-ship is zero total energy of space-ship ` = - (GMm)/( R) - (GM.m)/(R.) = - Gm (M/R + (M.)/(R.) )` `therefore `Energy required to rocket out the spaceship from the solar system = -(total energy of spaceship) ` = - [-Gm(M/R +(M.)/(R.) ) ] = Gm (M/R + (M.)/(R.) ) = 6.67 XX 10^(-11) xx 1000 xx [ (2 xx 10^30)/(2.28 xx 10^11) + (6.4 xx 10^23)/(3395 xx 10^3)]` `= 6.67 xx 10^(-8) [ (20)/(2.28) + (6.4)/(33.95) ] xx 10^18 J = 4.89 xx 10^11 J`