At what height above the surface of Earth is there a 1% difference between the approximate magnitude of gravitational field $\mathbf{9}\mathbf{.}\mathbf{8}\mathbf{}\frac{\mathbf{N}}{\mathbf{kg}\mathbf{}}$and the actual value of gravitational field at that location? This is, at what height y above the Earth’s surface is$\frac{\mathbf{G}{\mathbf{M}}_{\mathbf{e}}}{\mathbf{(}{\mathbf{R}}_{\mathbf{e}}\mathbf{+}\mathbf{y}{\mathbf{)}}^{\mathbf{2}}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{99}\frac{\mathbf{G}{\mathbf{M}}_{\mathbf{e}}}{{{\mathbf{R}}_{\mathbf{e}}}^{\mathbf{2}}}$

The given data can be listed below as,

The gravitational force formula defines the magnitude of the force between the two objects. It is also known as Newton’s law of gravitation.

Gravitational force = $\frac{(\mathrm{Gravitational}\mathrm{constant})(\mathrm{mass}\mathrm{of}\mathrm{object}1)(\mathrm{mass}\mathrm{of}\mathrm{object}2)}{(\mathrm{distance}\mathrm{between}\mathrm{the}\mathrm{two}\mathrm{objects})}$

The value of G at a height “h” above the earth’s surface from the gravitational law of motion can be expressed as,

$g_h=\frac{G{M}_e}{(R_e+h{)}^2}$

Here, G is the gravitational constant,$M_e$ is the mass of the earth,$R_e$ is the radius of Earth.

Rewriting the above equation,

$$\begin{gathered} g_h=\frac{G{M}_e}{R_e{\left(1+{\displaystyle \frac{h}{R_e}}\right)}^2} \\ {g}_h=\frac{g_s}{{\left(1+\frac{h}{R_e}\right)}^2} \end{gathered}$$

for$h<<R_e$

role="math" localid="1658907029410" $$\begin{gathered} g_h=g_s{\left(1+\frac{h}{R_e}\right)}^{-2} \\ =g_s\left(1-\frac{2h}{R_e}\right) \\ {g}_h=g_s\left(1-\frac{2h}{R_e}\right) \\ {g}_h=g_s\left(1-\frac{1}{100}\right) \\ {g}_h=0.99g_s \end{gathered}$$

$0.99=1-\frac{2h}{R_e}$

Now, $R_e=6400km$

Substitute all the values in the above equation.

$$\begin{gathered} \frac{2h}{R_e}=0.01 \\ h=\frac{0.01\times 6400}{2} \\ h=32km \end{gathered}$$

Thus, at height 32 km above the surface of Earth, there is a 1% difference between the approximate magnitude of gravitational field and actual magnitude of gravitational field at that location