Gravitation

JEE Main level, single correct, medium

173 of 223

Question 173 of 223

A space ship of mass 2×104kg2\times {10}^{4}kg is launched into a circular orbit close to the earth surface. The additional velocity to be imparted to the space ship in the orbit to overcome the gravitational pull will be (if g=10ms−2g=10m{s}^{-2} and radius of earth =6400km=6400km ):

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Answer: (B) 8(2−1)kms−18(\sqrt{2}-1)km{s}^{-1}

The orbital velocity of satellite is
vorb=GMR{v}_{orb}=\sqrt{\dfrac{GM}{R}}
The escape velocity is
vesc=2GMR{v}_{esc}=\sqrt{\dfrac{2GM}{R}}
Hence, the additional velocity required is
Δv=vesc−vorb=2GMR−GMR=(2−1)GMR\Delta v={v}_{esc}-{v}_{orb}=\sqrt{\dfrac{2GM}{R}}-\sqrt{\dfrac{GM}{R}}=(\sqrt{2}-1)\sqrt{\dfrac{GM}{R}}
=(2−1)gR=(\sqrt{2}-1)\sqrt{gR}
=(2−1)10×6400×103=(\sqrt{2}-1)\sqrt{10\times 6400\times {10}^{3}}
=8(2−1)kms−1=8(\sqrt{2}-1)km{s}^{-1}

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