A particle \hat{p} is sliding down a frictionless hemispherical bowl. It passes the point A. at \{ = 0 . At this instant of time, the horizontal component of its velocity is v. A bead $\hat{q}$ of the same mass as \hat{p} is ejected from \hat{A} at \{ = 0 along the horizontal string \(A\beta\) (see figure) with the speed \vec{v} . Friction between the bead and the string may be neglected. Let t_p and \(\mathcal{E}_{c}\) be the respective time taken by \(\mathcal{P}\) and $\theta$ to reach the point \(\beta\) . Then