When people on a cruise ship cross Earth\'s equator for the first time, they are

Question

When people on a cruise ship cross Earth's equator for the first time, they are awarded a certificate to commemorate the experience. Imagine that you are the tour director for a trip to the Moon. Transplanetary Tours promises tour participants a certificate to commemorate their passage from the stronger influence of Earth's gravitational pull to the stronger pull of the Moon at the point where the two forces on your spaceship are equal.

Where on the trip should you award the certificate?
Answer should be in km from Earth.

Explanation / Answer

gravitational pull on space craft due to earth = gravitational pull on space craft due to Moon GMm / x^ 2 = GM ' m / ( r-x)^ 2                       M / x^ 2 = M ' / ( r-x) ^ 2                  [(r-x)/x]^ 2 = M ' / M         (r-x) / x = [M ' / M ]    ---------------(1) where M = mass of the earth = 5.98 * 10 ^ 24 kg           M ' =Mass of Moon = 7.36*10^22 kg            x = required distance             r = distance between earth and Moon               = 3,84440 km plug the values in eq( 1) you get , (r-x) / x = 0.1109    r-x = 0.1109 x 1.1109 x = r              x = r /1 .1109                 = ( 384440 km ) / 1.1109                 = 346049.27 km

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