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# A partical is projected along a horizontal field whose co-efficient of friction varies $\mu = \large\frac{A}{x^2}$ where x is a distance from origin in meters and A is a positive constant. The initial distance of particle is 1m from orgin and its velocity is radially outwards. The minumum initial velocity at this point so that the particle never stops is

$(a)\;\sqrt {2gA} \quad (b)\;\infty \quad (c)\; 2 \sqrt {gA} \quad (d)\;4 \sqrt {gA}$

The body must start from $x=1$.Then, the work done against friction must be equal to the initial kinetic energy given to the body in the form of initial velocity of the body
$\large\frac{1}{2}$$mv^2=\int \limits_1^{\infty} \mu mg \;dx \large\frac{v^2}{2}$$=Ag \int \limits_1^{\infty} \large\frac{1}{x^2}$$dx \large\frac{v^2}{2}$$=Ag \bigg[ \large\frac{-1}{x}\bigg] _1 ^{\infty}$
$v^2=2 Ag$
$v=\sqrt {2gA}$
Hence a is the correct answer.
edited Jun 13, 2014

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