a
Angular momentum conservation about A

\(mv _{0} a \cos \theta+\frac{2}{5} ma ^{2} \omega\)

\(= mva +\frac{2}{5} ma ^{2} \omega^{1}\)

\(mv _{0} a \left[\frac{2}{5}+\cos \theta\right]=\frac{7}{5} mva\)

\(v =\frac{5}{7}= v _{0}\left[\frac{2}{5}+\cos \theta\right]\)

\(\frac{1}{2} mv ^{2}+\frac{1}{2} I \omega^{2}=\frac{7}{10} mv ^{2}= mgh\)

\(d=\frac{10 v_{0}^{2}}{7 g \sin \theta}\)