A) \[\sqrt{2}{{u}_{es.}}\]
B) \[\frac{{{u}_{es}}}{\sqrt{2}}\]
C) \[2{{u}_{es}}\]
D) \[{{4}_{ue}}\]
Correct Answer: A
Solution :
Escape velocity \[{{\upsilon }_{es}}k=\sqrt{\frac{2GM}{R}}\] \[=\sqrt{\frac{2G\frac{4}{3}\pi {{R}^{3}}d}{R}}=R\sqrt{\frac{8}{3}\pi Gd}\] \[\frac{{{\upsilon }_{e{{s}_{1}}}}}{{{\upsilon }_{e{{s}_{2}}}}}=\frac{{{R}_{1}}}{{{R}_{2}}}\sqrt{\frac{{{d}_{1}}}{{{d}_{2}}}}\] \[\frac{{{\upsilon }_{es}}}{{{\upsilon }_{e{{s}_{2}}}}}=\frac{{{R}_{1}}}{2{{R}_{1}}}\sqrt{\frac{d}{d/2}}\] \[=\frac{1}{\sqrt{2}}\] \[\Rightarrow \] \[{{\upsilon }_{e{{s}_{2}}}}=\sqrt{2}{{\upsilon }_{es}}\]
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