How far above the Earth’s surface will the acceleration of gravity by half what it is at the surface?

The relation between the distance from Earth's surface and radius of Earth can be determined by using the equation of acceleration due to gravity. Hence the distance from the Earth's surface is the function of Earth's radius.

The acceleration due to gravity is given by the relation:

\(g = \frac{{G{M_{{\rm{earth}}}}}}{{{r^2}}}\)

Here, \(r\) is the distance from the centre of the earth to the location.

The location where \(g = \frac{1}{2}{g_{{\rm{surface}}}}\) can be given as,

\(\begin{aligned}\frac{{G{M_{{\rm{earth}}}}}}{{{r^2}}} &= \frac{1}{2}\frac{{G{M_{{\rm{earth}}}}}}{{R_{{\rm{earth}}}^2}}\\r &= \sqrt 2 {R_{{\rm{earth}}}}\end{aligned}\)

Substitute the values in the above equation,

\(\begin{aligned}r - {R_{{\rm{earth}}}} &= \left( {\sqrt 2 - 1} \right){R_{{\rm{earth}}}}\\ &= \left( {\sqrt 2 - 1} \right){R_{{\rm{earth}}}}\\ &= {\rm{0}}{\rm{.414}}\;{{\rm{R}}_{{\rm{earth}}}}\end{aligned}\)

Thus, the required distance is \({\rm{0}}{\rm{.414}}\;{{\rm{R}}_{{\rm{earth}}}}\).