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7.Gravitation
hard
What is 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 altitute of $2R$ ?
A
$\frac {5GmM}{6R}$
B
$\frac {2GmM}{3R}$
C
$\frac {GmM}{2R}$
D
$\frac {GmM}{3R}$
Solution
By conservation of Energy.
$-\frac{\mathrm{GMm}}{\mathrm{R}}+\mathrm{E}=-\frac{\mathrm{GMm}}{3 \mathrm{R}}+\frac{1}{2} \mathrm{mv}_{0}^{2}$
$\mathrm{E}=-\frac{\mathrm{GMm}}{3 \mathrm{R}}+\frac{1}{2} \mathrm{m}\left(\frac{\mathrm{GM}}{3 \mathrm{R}}\right)+\frac{\mathrm{GMm}}{\mathrm{R}}$
$E=-\frac{G M m}{3 R}+\frac{G M m}{6 R}+\frac{G M m}{R}$
$\therefore \quad \mathrm{E}=\frac{5 \mathrm{GMm}}{6 \mathrm{R}}$
Standard 11
Physics
