Velocity at mean position of a particle executing $S.H.M.$ is $v$, they velocity of the particle at a distance equal to half of the amplitude
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(c) Velocity in mean position $v = a\omega ,$ velocity at a distance of half amplitude.
$v' = \omega \sqrt {{a^2} - {y^2}} $
$v' = \omega \sqrt {{a^2} - {y^2}} $
$ = \omega \sqrt {{a^2} - \frac{{{a^2}}}{4}} $
$ = \sqrt {\frac{3}{2}} \,a\omega $
$= \sqrt {\frac{3}{2}} \,v$
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