Consider a disc rotating in the horizontal plane with a constant angular speed $\omega$ about its centre $O.$ The disc has a shaded region on one side of the diameter and an unshaded region on the other side as shown in the figure. When the disc is in the orientation as shown, two pebbles $P$ and $Q$ are simultaneously projected at an angle towards $R.$ The velocity of projection in the $y-z$ plane and is the same for both pebbles with respect to the disc. Assume that

$\left(\mathrm{i}\right)$ they land back on the disc before the disc has completed $\frac{1}{8}$ rotation,

$\left(\mathrm{ii}\right)$ their range is less than half the disc radius, and

$\left(\mathrm{iii}\right)$ $\omega$ remains constant throughout. Then