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.

Explanation Video

Derivation of the formulas for Energies of Satellites in Orbit

Interesting

SpaceX Launch

Concept Test

1269

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