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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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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 knowÂ Â , $\frac{m_{e}}{m_{m}} = 81.3$ and $r_{e}+r_{m} = 378000 km$

so, $r_{e} = 340262 km$

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