The minimum energy required to launch a satellite of mass $m$ from the surface of a planet of mass M and…

The minimum energy required to launch a satellite of mass $m$ from the surface of a planet of mass M and radius R in a circular orbit at an altitude of $2 R$ is
  1. $\frac{5 \mathrm{GMm}}{6 \mathrm{R}}$
  2. $\frac{2 \mathrm{GMm}}{3 \mathrm{R}}$
  3. $\frac{\mathrm{GMm}}{2 \mathrm{R}}$
  4. $\frac{\mathrm{GMm}}{3 \mathrm{R}}$

Solution

Orbital energy, $\mathrm{E}_0=\frac{-\mathrm{GMm}}{2(\mathrm{R}+\mathrm{h})}$ $\therefore \quad \mathrm{E}_0=\frac{-\mathrm{GMm}}{2(\mathrm{R}+2 \mathrm{R})}=\frac{-\mathrm{GMm}}{6 \mathrm{R}} \quad \ldots(\because \mathrm{~h}=2 \mathrm{R})$
Energy at surface $E=\frac{-\mathrm{GMm}}{R}$ $\begin{aligned} \therefore \quad \text { Min. energy required } & =\mathrm{E}_0-\mathrm{E} \\ & =\frac{-\mathrm{GMm}}{6 \mathrm{R}}-\left(\frac{-\mathrm{GMm}}{\mathrm{R}}\right) \\ & =\frac{5 \mathrm{GMm}}{6 \mathrm{R}} \end{aligned}$

Asked in: MHT CET 2024 (15 May Shift 2)

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