\begin{aligned}
&\text { Average velocity }=\frac{\text { total distance travelled }}{\text { total time taken }} \\
&\text { Average velocity }=\frac{S}{t} \ldots(1) \\
&\text { Average velocity can also be written as } \frac{u+v}{2} \\
&\text { Average velocity }=\frac{u+v}{2} \ldots(2) \\
&\text { From equations (1) and (2) } \\
&\frac{S}{t}=\frac{u+v}{2} \quad \ldots .(3) \\
&\text { The first equation of motion is } v=u+a t . \\
&\text { Substituting the value of } v \text { in equation (3), we get } \\
&\frac{S}{t}=\frac{u+u+\text { at }}{2}=\frac{(u+u+\text { at }) t}{2} \\
&=\frac{(2 u+\text { at }) t}{2}=\frac{2 u t+\text { at }^{2}}{2}=\frac{2 u t}{2}+\frac{\text { at }^{2}}{2} \\
&S=u t+\frac{1}{2} \text { at }^{2} \\
&S=u t+\frac{1}{2} \text { at }^{2} \\
&\text { II equation of motion. }
\end{aligned}
