The velocity $v$ and displacement $x$ of a particle executing simple harmonic motion are related as

Question

The velocity $v$ and displacement $x$ of a particle executing simple harmonic motion are related as $v \frac{d v}{d x}=-\omega^2 x$. If at $x=0, v=v_0$. The velocity $v$ when the displacement becomes $x$ is

Professor PC Thomas Classes · Answer

Integrating $v dv = -\omega^2 x dx$ gives $\frac{v^2}{2} = -\frac{\omega^2 x^2}{2} + C$. Using initial conditions yields $v = \sqrt{v_0^2 - \omega^2 x^2}$.