# From the Earth to the Moon

Jules Verne's classic science fiction story describes a cannon firing a projectile (with three people inside!) at the Moon. The story says that the projectile was fired so that it just reached the neutral zone (where the gravitational pull of the Moon on the projectile equals that of the Earth, called an unstable equilibrium point) when the Moon was at its closest point to the Earth in its orbit. The Moon's gravitational attraction then dominates and the projectile is captured by the Moon. If it is fired at a speed that just lets it pass through the neutral zone it reaches the Moon approximately 7 days after leaving the Earth. As the concepts in this problem are in the Space section of the NSW HSC Physics course, an interesting exercise is to apply the gravitational potential energy equation to this problem. Given the following data:

closest approach distance of the centre of the Moon to the centre of the Earth: 363,100 km
radius of the Earth: 6.38x106m,
radius of the Moon: 1.74x106m,
mass of the Earth:5.97x1024kg,
mass of the Moon:7.36x1022kg.

(i) What is the least speed at which the shell can be projected from the Earth if it is to reach the equilibrium point between the Earth and Moon? Neglect the orbital and rotational motions of each object.

(ii) If the projectile is slightly pushed from the equilibrium point towards the Moon, with what speed does it strike the Moon? Neglect the orbital and rotational motions of each object. (In Verne's story rockets were fired that put the shell in a lunar orbit)

(ii) If the orbital movement of the Moon is included, is the projection speed greater or less than in (i)?

greater, see K R Symon, "Mechanics", page 292

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