The total time of journey

\(-s=u t-\frac{1}{2} g t^2\)

\(\Rightarrow H=\frac{1}{2} g T^2 \quad \ldots(i)\)

\(\frac{-H}{2}=u t-\frac{1}{2} g t^2 \Rightarrow T=\sqrt{\frac{2 H}{g}}\)

\(\Rightarrow \frac{H}{2}=\frac{1}{2} g t^2\)

\(\Rightarrow \frac{1}{2} g T^2=g t^2 \quad(\because u t=0)\)

\(\Rightarrow t=\frac{T}{\sqrt{2}}\)

\(\Rightarrow\) Second half time \(=T-t=T-\frac{T}{\sqrt{2}}=T\left(1-\frac{1}{\sqrt{2}}\right)=\sqrt{\frac{2 H}{g}}\left(1-\frac{1}{\sqrt{2}}\right)=\sqrt{\frac{H}{g}}(\sqrt{2}-1)\)