Lets set the origin at the Earth. Then the force at distance r from the Earth is given as follows:
F1=Gr2MEmwhere G=6.67×10−11kg⋅s2m3 is the gravitational constant, ME is the mass of the Earth, m is the mass of the ship.
The force from the Moon at the same point is given as follows:
F2=G(d−r)2MMm where d is the distance between the center of the Earth and the center of the Moon. The total gravitational force is 0 if:
F1=F2Gr2MEm=G(d−r)2MMmr2ME=(d−r)2MM Expressing r, find:
(ME−MM)r2−2dMEr+MEd2=0 Substituting the values and solving the quadratic equation, obtain:
r≈2.8×108m
Answer. 2.8×108m.
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