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EEJ MAIN Physics QUESTION #6998
Question 1

What is the minimum energy needed to launch a satellite of mass $m$ from the surface of a planet of mass $M$ and radius $R$ into a circular orbit at altitude $2R$?

  • $\dfrac{5GmM}{6R}$✔️
  • $\dfrac{2GmM}{3R}$
  • $\dfrac{GmM}{2R}$
  • $\dfrac{GmM}{3R}$
Correct Answer Explanation

The orbit radius is $r = R + 2R = 3R$.

Orbital KE: $KE = \frac{GmM}{2r} = \frac{GmM}{6R}$

Total mechanical energy in orbit: $E_{orbit} = -\frac{GmM}{2r} = -\frac{GmM}{6R}$

Energy on surface: $E_{surface} = -\frac{GmM}{R}$

Minimum energy required:

$\Delta E = E_{orbit} - E_{surface} = -\frac{GmM}{6R} + \frac{GmM}{R} = \frac{GmM}{R}\left(1 - \frac{1}{6}\right) = \frac{5GmM}{6R}$