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  4. Escape velocity from Earth is about 11 km/s. The escape velocity from a planet, 4 times the size of the Earth with same mean density is

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Q. Escape velocity from Earth is about 11 km/s. The escape velocity from a planet, 4 times the size of the Earth with same mean density is

COMEDKCOMEDK 2007Gravitation

Solution:

Escape velocity of a planet
$v=\sqrt{\frac{2GM}{R}} = \sqrt{\frac{2G \times\frac{4}{3} \pi R^{3} \times\rho}{R}}$
$v=R\sqrt{\frac{8\pi}{3} G\rho}$
For same $\rho, v \propto R$
$\therefore \:\: \frac{v_{E}}{v_{P}} = \frac{R_{E}}{R_{P}} $
Here, $v_{E} = 11 Km s^{-1} , R_{E} = R, R_{p} = 4R, v_{P } = ? $
$ \therefore \:\:\: \frac{11}{v_{P} } = \frac{R}{4R}$
or, $ v_{P} = 44 km s^{-1}$