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

Answer 1

Answer: (D)

We know
$V _{0}=\sqrt{\frac{ GM }{ r }} \& g =\frac{ GM }{ R ^{2}}$
From energy conservation
$U _{ i }+ K _{ i }= U _{ f }+ K _{ f }$
$-\frac{ GMm }{ R }+ K _{ i }=-\frac{ GMm }{2 R }+\frac{1}{2} mv _{0}^{2}$
$K _{ i }=\frac{ GMm }{2 R }+\frac{1}{2} m \left(\sqrt{\frac{ GM }{2 R }}\right)^{2} \Rightarrow K _{ i }=\frac{3 GMm }{4 R } \Rightarrow K _{ i }=\frac{3}{4} mgR$