7.3.4 Energy of Satellites

Consider a satellite of mass m in a circular orbit of radius r around the Earth. Since the orbital speed depends on the orbital radius, the KE of the satellite is also a function of r.

\displaystyle \begin{aligned}({{F}_{{net}}}&=ma)\\\frac{{GMm}}{{{{r}^{2}}}}&=\frac{{m{{v}^{2}}}}{r}\\\frac{1}{2}\frac{{GMm}}{r}&=\frac{1}{2}m{{v}^{2}}\\KE&=\frac{1}{2}\frac{{GMm}}{r}\end{aligned}

Needless to say, the GPE of the satellite is also a function of r.

\displaystyle GPE=-\frac{{GMm}}{r}

Finally the total energy of the satellite is also a function of r.

\displaystyle \begin{aligned}TE&=KE+GPE\\&=(\frac{1}{2}\frac{{GMm}}{r})+(-\frac{{GMm}}{r})\\&=-\frac{1}{2}\frac{{GMm}}{r}\end{aligned}

We can now sketch the graphs of the variation of a satellite’s KE, GPE and TE with orbital radius r.

Some important points to note:

  • At any r, GPE=-2KE=2TE
  • TE and GPE increase with r, whereas KE decreases with r.
  • To move a satellite from a lower orbit to a higher orbit, we must increase the TE of the satellite. This requires the burning of rocket fuel so that the propulsion force does (positive) work on the satellite.

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