Problem 4:

An alien spacecraft is hiding behind the moon, such that the Earth, the Moon, and the spacecraft are all in a straight line, with the Moon somewhere between the Earth and the spacecraft. Calculate the distance r between the Moon and the spacecraft such that the Earth and the Moon exert equal forces on the alien (Note: this is not the same as the homework problem where the object was between the Earth and the Moon).

Earth-Moon distance   R_EM = 3.84 x 10⁸ m
Mass of the Earth   M_E = 5.98 x 10²⁴ kg
Mass of the Moon   M_m = 7.26 x 10²² kg

Solution:

    F =  G M_E m / (R_EM + r)² = G M_m m / r²

cancel factors of G and m, and rearrange to yield

     r²/(R_EM + r)² = M_m/M_E

This is a quadratic equation in r, which can be solved various ways.
Here's an easy way: take square root of both sides

      r/(R_EM + r) = [M_m/M_E]^½

Call    [M_m/M_E]^½ = C = 0.1102.    Now rearrange further

     r  =  R_EMC + rC

     r = R_EMC/(1-C)  =  (3.84 x 10⁸ m)(0.1102)/(1-0.1102)  =  4.76 x 10⁷ m

Test 3
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