Question

A satellite of mass m, initially at rest on the earth, is launched into a circular orbit at a height equal to the radius of the earth. The minimum energy required is:

• A. $\frac{\sqrt{3}}{4}mgR$
• B. $\frac{1}{2}mgR$
• C. $\frac{1}{4}mgR$
• D. $\frac{3}{4}mgR$

Answer 1

The correct option is D $\frac{3}{4}mgR$

Total Energy in the circular orbit:

$TE=\frac{1}{2}m{v}^2-\frac{gmM}{2R}\because \frac{m{v}^2}{2R}=\frac{GMm}{{\left(2R\right)}^2}TE=\frac{-1}{4}\frac{GMm}{R}Changeinenergy=\frac{-1}{4}\frac{GMm}{R}-(-\frac{GMm}{R})=\frac{3}{4}\frac{GMm}{R}$