Derive the equations $\mathrm{s}=\mathrm{ut}+\frac{1}{2}{\mathrm{t}}^2$ using graphical method.

Consider a body moves with an initial velocity = u
Uniform acceleration = a
Final velocity = v ; Time taken = t

For this motion v – t graph is plotted as,

S = Area of OABD
$\mathrm{S}=\text{ Area of □}\mathrm{OACD}+\text{ Area of }\mathrm{△}\mathrm{ABC}$
$\mathrm{S}=(\mathrm{OA}\times \mathrm{OD})+\frac{1}{2}(\mathrm{AC}\times \mathrm{BC})$
$\mathrm{S}=(\mathrm{u}\times \mathrm{t})+\frac{1}{2}[\mathrm{t}\times (\mathrm{v}-\mathrm{u}\left)\right] [\because \mathrm{v}-\mathrm{u}=\mathrm{at}]$
$\mathrm{S}=\mathrm{ut}+\frac{1}{2}[\mathrm{t}\times \mathrm{at}]\Rightarrow \mathrm{S}=\mathrm{ut}+\frac{1}{2}{\mathrm{at}}^2$