Explanation:

When the ring is at the maximum height, the wedge and the ring have the same horizontal component of velocity. As all the surfaces are smooth, in the absence of friction between the ring and wedge and surface angular velocity of the ring remains constant.
From conservation of mechanical energy
\(\frac{1}{2} M v^{2}+\frac{1}{2} I \omega^{2}=\frac{1}{2} M v_{1}^{2}+\frac{1}{2} I \omega^{2}+\frac{1}{2} M v_{1}^{2}+m g h\) where \(\mathrm{v}_{1}=\) final common velocity \(=\frac{\mathrm{v}}{2}\)
\(h=\frac{\mathrm{v}^{2}}{4 g}\)