A body is projected vertically up with u. Its velocity at half its maximum height is

Maximum height for a vertically projected body is 

$\mathrm{H} =\frac{{\mathrm{u}}^2}{2\mathrm{g}}$

From v² - u² = 2as
$\begin{array}{l}{\mathrm{v}}^2-{\mathrm{u}}^2=-2\mathrm{g}\frac{\mathrm{H}}{2}\\ {\mathrm{v}}^2={\mathrm{u}}^2-\mathrm{g}\frac{{\mathrm{u}}^2}{2\mathrm{g}}\\ {\mathrm{v}}^2=\frac{{\mathrm{u}}^2}{2}\Rightarrow \mathrm{v}=\frac{\mathrm{u}}{\sqrt{2}}\end{array}$