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Show the nature of the following graph for a satellite orbiting the earth.
$(a)$ $KE$ versus orbital radius $R$
$(b)$ $PE$ versus orbital radius $R$
$(c)$ $TE$ versus orbital radius $R$
Solution
consider the diagram, where a satellite of mass $m$, moving around the earth in a circular orbit of radius $\mathrm{R}$.
Orbital speed of the satellite orbiting the earth is given by $\mathrm{V}_{0}=\sqrt{\frac{\mathrm{GM}}{\mathrm{R}}}$
where, $M$ and $R$ are the mass and radius of the earth.
$(a)$ $\therefore \mathrm{KE}$ of a satellite of mass $m$,
$\mathrm{K}=\frac{1}{2} m v_{0}^{2}=\frac{1}{2} m \times \frac{\mathrm{GM}}{\mathrm{R}} $
$\therefore\mathrm{K} \propto \frac{1}{\mathrm{R}}$It means the $KE$ decreases exponentially with radius. The graph for $KE$ versus orbital radius $\mathrm{R}$ is shown in figure
