Sol. Total time of flight $=\mathrm{T}$ $$ T=\sqrt{\frac{2 h}{g}}+\sqrt{\frac{2(H-h)}{g}} $$ For max. time $=\frac{\mathrm{dT}}{\mathrm{dh}}=0$ $$ \begin{aligned} & \sqrt{\frac{2}{g}}\left(\frac{-1}{2 \sqrt{\mathrm{H}-\mathrm{h}}}+\frac{1}{2 \sqrt{\mathrm{~h}}}\right)=0 \\ & \sqrt{\mathrm{H}-\mathrm{h}}=\sqrt{\mathrm{h}} \\ & \mathrm{~h}=\frac{\mathrm{H}}{2} \Rightarrow \frac{\mathrm{H}}{\mathrm{~h}}=2 \end{aligned} $$