# At what distance from the Earth will the pull of the Moon on the spaceship exceed the pull of the Earth?A spaceship is launched and starts moving directly towards the Moon. At what distance from...

At what distance from the Earth will the pull of the Moon on the spaceship exceed the pull of the Earth?

A spaceship is launched and starts moving directly towards the Moon. At what distance from the Earth will the pull of the Moon on the spaceship exceed the pull of the Earth? Ignore the effect of the Sun in this calculation.

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Let the mass of the spaceship be m.

And let the space ship be at a distance of d from the earth.

Then the force exerted by the earth on the space is given by:

Fe = G*M1*m/d^2, where M1is thw mass of the earth.

The distance between the moon and the space ship is D-d , where D is the distance between the earth and moon.

So the force F2 of moon exerted on the space ship is given by:

F2 = G*M2*m/(D-d)^2..

Since moon's force F2 > earth's force F1 at d, we get the relation:

GM2*m/(D-d)^2 >G*M1*m/d^2. We cancel the comon factors, gm.

=> M2*d^2 > M1(D^2-2Dd+d^2).

=> M1(D^2-2Dd+d^2)-M2d^2 < 0.

=> d^2(M1-M2) -2M1D*d +M1D^2 < 0.

The zeros of the expression on the left are d1 = {2M1*D+sqrt(4M1D^2-4(M1-M2)*M1*D^2)}/2(M1-M2)

d1 = {M1D+Dsqrt(M1M2)}/(M1-M2).

d2 = {M1D-Dsqrt(M1M2)}/(M1-M2).

So d > d2 = D{M1-sqrt(M1M2)}/(M1-M2) ....(1)

Put the values M1 = 5.978*10^24 kg , M2 = 7.353*10^22 kg and D = 3.844*10^8 meter. and we get: d > 0.9001638D = 3.46*10^8 meter.

Therefore when the space ship is father than 3.46*10^8 meter from earth the moon's gravitational force is more than that of the earth.