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When a spacecraft travels from the Earth to the Moon, both the Earth and the Moon exert a gravitational force on the spacecraft. Eventually, the spacecraft reaches a point where the Moon's gravitational attaction overcomes the Earth's gravity. How far from the Earth's surface must the spacecraft be for the gravitational forces from the Moon and the Earth to just cancel?
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Added Thu, 06 Aug '15
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Force due to gravity is given by

F =\frac{Gm_{1}m_{2}}{r^{2}}

Let m_{e} be mass of earth and r_{e} be the distance of spacecraft fom earth

Similarly, Let m_{m} be mass of moon and r_{m} be the distance of spacecraft fom moon

For the forces to get cancelled

\frac{Gm_{e}m_{s}}{r_{e}^{2}} =\frac{Gm_{m}m_{s}}{r_{m}^{2}}

\frac{{r_{e}^{2}}}{{r_{m}^{2}}} =\frac{m_{e}}{m_{m}}

We know  , \frac{m_{e}}{m_{m}} = 81.3 and r_{e}+r_{m} = 378000 km

so, r_{e} = 340262 km

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