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

Answer 1

Answer: (D)

We know
$V_0=\sqrt{\frac{GM}{r}}\&g=\frac{GM}{R^2}$
From energy conservation
$U_i+K_i=U_f+K_f$
$-\frac{GMm}{R}+K_i=-\frac{GMm}{2R}+\frac{1}{2}m{v}_0^2$
$K_i=\frac{GMm}{2R}+\frac{1}{2}m{\left(\sqrt{\frac{GM}{2R}}\right)}^2\Rightarrow K_i=\frac{3GMm}{4R}\Rightarrow K_i=\frac{3}{4}mgR$