A particle of mass m is thrown upwards from the surface of the earth, with a velocity u. The mass and the radius of the earth are, respectively, M and R.G are gravitational constant and g is acceleration due to gravity on the surface of the earth. The minimum value of u so that the particle does not return back to earth is

$square root of fraction numerator 2 GM over denominator 2 end fraction end root$
$square root of fraction numerator 2 GM over denominator straight R squared end fraction end root$
$square root of 2 gR squared end root$
$square root of fraction numerator 2 GM over denominator straight R squared end fraction end root$

Solution

square root of fraction numerator 2 GM over denominator 2 end fraction end root

straight v subscript straight e space equals space square root of 2 gR end root straight v subscript straight e space equals space square root of fraction numerator 2 GM over denominator straight R squared end fraction straight x space straight R end root space space left parenthesis therefore space GM space equals space gR squared right parenthesis straight v subscript straight e space equals space square root of fraction numerator 2 GM over denominator straight R end fraction end root

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