The velocity of a particle in simple harmonic motion at displacement $y$ from mean position is
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Let the equation of motion for the particle be, $y=$ Asinot
Differentiating this with respect to time will give us the velocity of the particle. $\frac{d y}{d t}=A \omega \cos \omega t$
We have, $\sin \omega t =\frac{y}{A}$ and therefore we can calculate $\cos \omega t =\frac{\sqrt{A^2-y^2}}{A}$ (Use Pythagoras theorem to calculate cosine from sine)
Use the value of cosine in equation $(1)$,
$V = A \omega \frac{\sqrt{A^2- y ^2}}{A}$
or, $v=\omega \sqrt{A^2-y^2}$ is our required answer.
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