A particle 𝑃 is sliding down a frictionless hemispherical bowl. It passes the point 𝐴 at 𝑡 = 0. At this instant of time, the horizontal component of its velocity 𝑣. A bead 𝑄 of the same mass as 𝑃 is ejected from 𝐴 to 𝑡 = 0 along the horizontal string 𝐴𝐵 (see figure) with the speed 𝑣. Friction between the bead and the string may be neglected. Let 𝑡𝑝 and 𝑡𝑄 be the respective time taken by 𝑃 and 𝑄 to reach the point 𝐵. Then