The velocity vector $v$ and displacement vector $x$ of a particle executing SHM are related as $\frac{v d v}{d x}=-\omega^2 x$ with the initial condition $v=v_0$ at $x=0$. The velocity $v$, when displacement is $x$, is
AIIMS 2015, Diffcult
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(b)
As it is $SHM$ so the equation of motion will be $F=- kx$ or $\frac{ vdv }{ dx }=-\omega^2 x$
Now integrating the expression with boundary condition, $\int \limits_{ v _0}^v v d v=-\omega^2 \int \limits_0^\pi xdx$
or $\frac{1}{2}\left[v^2-v_0^2\right]=-\frac{\omega^2 x^2}{2}$
or $v =\sqrt{ v _0^2-\omega^2 x ^2}$
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