Derive second equation of motion.

We know the first equation of motion:

$v=u+at$

Where v = final velocity, u = initial velocity, a = acceleration of a particle, and t = time taken to accelerate

Since the distance moved by a particle, S = average velocity $\times$time

Average velocity = $\frac{v+u}{2}$.

Therefore,

$S=\frac{v+u}{2}\times t$.

Using, the first equation of motion,

$v=u+at$

$$\begin{gathered} S=\frac{1}{2}\left(u+u+at\right)\times t \\ S=\frac{1}{2}(2u+at)\times t \\ S=ut+\frac{1}{2}a{t}^2 \end{gathered}$$

This is known as the second equation of motion.