Question

After watching ”War of the worlds” with Tom Cruise the aliens from Mars decided not to invade Earth but instead to leave the Solar system as soon as possible. They have decent space ships with propulsive jet velocity (ejection speed) 2000 m/s and dry mass (mass without fuel) of 10000 kg. Estimate how much fuel for each rocket is necessary to complete the mission. To increase efficiency the rockets are build in space on a high orbit so the Mars gravity may be neglected. The Solar mass is Ms = 2 E30 kg, gravitational constant is G = 6.7 E−11 N m2/kg2 , Mars orbit radius is rM = 2.3E11 m. Take into account that the Mars orbital motion can be used in a smart way and the fact that the rocket engine works for a short period of acceleration when the position of the rocket in Solar system does not significantly change (10 points)

Answer 1

The rocket propulsion equation is,

where

• is the change in velocity of the rocket without any external force.
• is the exhaust velocity.
• is the total initial mass of the rocket with fuel mass .
• is the dry mass of the rocket without fuel.

Now the modified form of the equation is,

we know the values of , . We can attack through orbital dynamics described below. So will be obtained.

Escape velocity is the velocity at which a particle can leave the gravitational bounds of an attractive larger mass.

Now the escape velocity of the Sun from a distance of mars is,

The orbital velocity of mars =

So here comes the smart part, any ship launching from mars will have the same velocity as Mars's orbital velocity. The whole planet will help the ship to reach the escape velocity. So the ship's = 34135.3 - 24137.3 = 9998 m/s.

So putting everything into modified rocket equation we have,