Question

A satellite is revolving around the earth. ratio of its orbital speed and escape speed will be?
1/√2
√2
√3
2√2​

Answer 1

$\blue{\bold{\underline{\underline{Answer:}}}}$

$\green{\tt{\therefore{v_{o}:v_{esc}=1:\sqrt{2}}}}$

$\orange{\bold{\underline{\underline{Step-by-step\:explanation:}}}}$

$\green{\underline \bold{Given :}} \\ \tt: \implies Satellite \: revolving \: around \: the \: earth \\ \\ \red{\underline \bold{To \: Find :}} \\ \tt: \implies \frac{Velocity \: of \: orbital \: speed( v_{o}) }{Escape \: velocity (v_{esc}) } =?$

• According to given question :

$\tt \circ \: m_{1} = Mass \: of \: earth \\ \\ \tt \circ \: m_{2} = Mass \: of \: sattelite \\ \\ \bold{As \: we \: know \: that} \\ \tt: \implies v_{o} = \sqrt{ \frac{G m_{1} m_{2}} {r} } \\ \\ \tt \circ \: m_{2} < < < m_{1} \\ \\ \tt \circ \: m_{1} m_{2} \approx m_{1} \\ \\ \tt: \implies v_{o} = \sqrt{ \frac{G m_{1} }{r} } - - - - - (1) \\ \\ \bold{As \: we \: know \: that} \\ \tt: \implies v_{esc} = \sqrt{ \frac{2Gm_{1} }{r} } - - - - - (2) \\ \\ \bold{For \: Ratio : } \\ \tt: \implies \frac{ v_{o} }{ v_{esc} } = \huge{\frac{ \sqrt{ \frac{Gm_{1} }{r} } }{ \sqrt{ \frac{2G m_{1} }{r} } }} \\ \\ \green{\tt: \implies \frac{ v_{o} }{ v_{esc} } = \frac{1}{ \sqrt{2} } } \\ \\ \green{\tt \therefore Ratio \: of \: v_{0} : v_{esc} \: is \: 1 : \sqrt{2} }$