The maximum height reached by the body projected vertically up with an initial velocity u is                                 

Given that

A body is project up vertically

Initial velocity$=\text{\hspace{0.17em}u}$

Final velocity$\text{v\hspace{0.17em}}=0,$$a=-\text{g}$      

$\text{t\hspace{0.17em}}=?$ $\text{h\hspace{0.17em}}=?$

We get$\text{v\hspace{0.17em}}=\text{\hspace{0.17em}u\hspace{0.17em}}-\text{\hspace{0.17em}gt}$  

$\text{o\hspace{0.17em}}=\text{\hspace{0.17em}u\hspace{0.17em}}-\text{\hspace{0.17em}gt}$ 

$\therefore {\text{\hspace{0.17em}\hspace{0.17em}t}}_{\text{ase}}=\text{\hspace{0.17em}u/2}$ 

We know$\text{h\hspace{0.17em}=\hspace{0.17em}ut\hspace{0.17em}}-\frac{1}{2}{\text{gt}}^{\text{2}}$   

$\text{h\hspace{0.17em}}=\text{\hspace{0.17em}u}{\left(\frac{\text{u}}{\text{g}}\right)}^{\text{2}}-\frac{\text{1}}{\text{2}}\text{\hspace{0.17em}g}{\left(\frac{\text{u}}{\text{g}}\right)}^{\text{2}}=\frac{{\text{u}}^{\text{2}}}{\text{2g}}$    

$\therefore {\text{\hspace{0.17em}\hspace{0.17em}h}}_{\text{max}}=\frac{{\text{u}}^{\text{2}}}{\text{2g}}$