A particle is describing simple harmonic motion. If its velocities are $v_{1}$ and $v_{2}$ when the displacements from the mean position are $y_{1}$ and $y_{2}$ respectively, then its time period is

In simple harmonic motion,

velocity $v = \omega \sqrt {A^{2} - y^{2}}$

$\therefore v_{1} = \omega \sqrt {A^{2} - y_{1}^{2}} \Rightarrow v_{1}^{2} = \omega^{2}A^{2} - \omega^{2}y_{1}^{2} .... (i)$

and $v_{2} = \omega \sqrt {A^{2} - y_{2}^{2}} \Rightarrow v_{2}^{2} = \omega^{2}A^{2} -\omega^{2} y_{2}^{2} ..... (ii)$

Solving equations (i) and (ii), we get

$v_{2}^{2} - v_{1}^{2} = \omega^{2} (y_{1}^{2} - y_{2}^{2})$

$\omega = \sqrt {\dfrac {v_{2}^{2} - v_{1}^{2}}{y_{1}^{2} - y_{2}^{2}}}$

$\Rightarrow T = \dfrac {2\pi}{\omega} = 2\pi \sqrt {\dfrac {y_{1}^{2} - y_{2}^{2}}{v_{1}^{2} - v_{2}^{2}}}$.