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 $=T-t=T-\frac{T}{\sqrt{2}}=T\left(1-\frac{1}{\sqrt{2}}\right)=\sqrt{\frac{2 H}{g}}\left(1-\frac{1}{\sqrt{2}}\right)=\sqrt{\frac{H}{g}}(\sqrt{2}-1)$