A) 8 km/s
B) 4 km/s
C) 6 km/s
D) 12 km/s
Correct Answer: A
Solution :
Escape velocity for earth is given by \[{{\upsilon }_{es(e)}}=\frac{\sqrt{2GMe}}{{{R}_{e}}}\] ?...(l) Escape velocity for planet is given by \[{{\upsilon }_{es(e)}}=\frac{\sqrt{2GMp}}{{{R}_{p}}}\] ...(ii) From equation (i) and (ii) we have \[\frac{{{\upsilon }_{es(p)}}}{{{\upsilon }_{es(e)}}}=\sqrt{\frac{{{M}_{p}}}{{{M}_{e}}}\times \frac{{{R}_{e}}}{{{R}_{p}}}}\] (Given, \[{{M}_{p}}=\frac{{{M}_{e}}}{4},{{R}_{p}}=\frac{{{R}_{e}}}{2}\]) So, \[\frac{{{\upsilon }_{es(p)}}}{{{\upsilon }_{es(e)}}}=\sqrt{\frac{{{M}_{e}}}{4\times {{M}_{e}}}\times \frac{2\times {{R}_{e}}}{{{R}_{e}}}}=\frac{1}{\sqrt{2}}\] So, \[{{\upsilon }_{es(p)}}=\frac{{{\upsilon }_{e}}}{\sqrt{2}}=\frac{11.2}{\sqrt{2}}=7.92\] \[=8km/s\]
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