The minimum energy required to launch a satellite of mass m from the surface of earth of mass M and radius R in a circul…

Initial energy (\({E}_{i}\)): The satellite is at rest on Earth’s surface (\(r=R\)), so

\({E}_{i}\text{  }=\text{  }{U}_{i}\text{  }=\text{  }−\text{ }\frac{G\text{ }M\text{ }m}{R}\mathrm{.}\)

Final energy (\({E}_{f}\)): The satellite is in a circular orbit at radius \(r=3R\) (Earth’s radius \(R\) plus altitude \(2R\)). For a circular orbit,

\({E}_{f}\text{  }=\text{  }−\text{ }\frac{G\text{ }M\text{ }m}{2\text{ }(3R)}\text{  }=\text{  }−\text{ }\frac{G\text{ }M\text{ }m}{6\text{ }R}\mathrm{.}\)

Minimum energy required = \(\Delta E={E}_{f}−{E}_{i}\):

\(\Delta E\text{  }=\text{  }−\text{ }\frac{G\text{ }M\text{ }m}{6\text{ }R}\text{  }−\text{  }(−\text{ }\frac{G\text{ }M\text{ }m}{R})\text{  }=\text{  }\frac{G\text{ }M\text{ }m}{R}(1\text{  }−\text{  }\frac{1}{6})\text{  }=\text{  }\frac{5\text{ }G\text{ }M\text{ }m}{6\text{ }R}\mathrm{.}\)

\(\text{Required energy  }=\text{  }\frac{5\text{ }G\text{ }M\text{ }m}{6\text{ }R}\mathrm{.}\)