a
Total time of flight \(=\mathrm{T}\)

\(T=\sqrt{\frac{2 h}{g}}+\sqrt{\frac{2(H-h)}{g}}\)

For \(\max\) time \(=\frac{d T}{d h}=0\)

\(\sqrt{\frac{2}{g}}\left(\frac{-1}{2 \sqrt{\mathrm{H}-\mathrm{h}}}+\frac{1}{2 \sqrt{\mathrm{h}}}\right)=0\)

\(\sqrt{\mathrm{H}-\mathrm{h}}=\sqrt{\mathrm{h}}\)

\(\mathrm{h}=\frac{\mathrm{H}}{2} \Rightarrow \frac{\mathrm{H}}{\mathrm{h}}=2\)