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#### From the moon to the earth

A rocket taking off from the surface of a planet or other celestial body will escape its gravitational field and continue moving in space, only if it reaches a speed that exceeds the “escape velocity”. Then, even without further propulsion, the rocket will keep moving indefinitely in space, unless it fires again, comes near the gravitational field of another celestial body, is hit by an asteroid or space junk, etc. For the earth’s moon, the escape velocity is 2.4 km/s.

Consider a spaceship that has landed on the surface of the moon and needs to return to earth. The first step would be to fire a rocket on the spaceship and bring it to a speed that is at least equal to the escape velocity. The mass of the spaceship and rocket with empty tanks (namely, without any fuel and oxidizer) is 40 kg. The mass flow rate of the exhaust gas from the rocket’s nozzle is controlled at 40 kg/s. The density of the exhaust gas is 0.50 kg/m3 and the exit diameter of the rocket nozzle is 320 mm. The gravitational acceleration on the moon’s surface is 1.6 m/s2 and its variation may be neglected over the distance that the spaceship travels while the rocket is fired. As you know, the earth’s moon has no atmosphere (unlike some moons of other planets). Determine the minimum mass of fuel/oxidizer that the spaceship must carry in order to escape the moon and the time required for the spaceship to reach the escape velocity. Also determine the height that the spaceship would reach when it attains the escape velocity.

Contributed by Stavros Tavoularis, Department of Mechanical Engineering, University of Ottawa, Ottawa, Canada. Image from NASA

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