Gravitation

JEE Main level, single correct, medium

130 of 223

Question 130 of 223

The escape velocity of a body on a planet AA is 12kms−112km{s}^{-1}. The escape velocity of the body on another planet BB, whose density is four times and radius is half of the planet AA, is

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Answer: (A) 12kms−112km{s}^{-1}

We know that escape velocity is given by,
v=2GMR⇒vAvB=MAMB×RBRA=ρARA3ρBRB3×RBRA=ρARA2ρBRB2=14×41⇒vB=vA=12kms−1v=\sqrt{\dfrac{2GM}{R}} \Rightarrow \dfrac{{v}_{A}}{{v}_{B}}=\sqrt{\dfrac{{M}_{A}}{{M}_{B}}\times \dfrac{{R}_{B}}{{R}_{A}}}=\sqrt{\dfrac{{\rho }_{A}{{R}_{A}}^{3}}{{\rho }_{B}{{R}_{B}}^{3}}\times \dfrac{{R}_{B}}{{R}_{A}}}=\sqrt{\dfrac{{\rho }_{A}{{R}_{A}}^{2}}{{\rho }_{B}{{R}_{B}}^{2}}}=\sqrt{\dfrac{1}{4}\times \dfrac{4}{1}} \Rightarrow {v}_{B}={v}_{A}=12km{s}^{-1}

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