Question

Derive the following equation for a uniformly accelerated motion, where the symbols have their usual meanings:

$s=ut+\frac{1}{2}a{t}^2$

Answer 1

Suppose, the distance traveled by the body in time $t$ is $s$.
For a uniformly accelerated motion, $\text{Average Velocity}=\frac{v+u}{2}$
Displacement $s=\text{Average Velocity}\times \text{Time}$
$s=\frac{v+u}{2}\times t.............\left(i\right)$

From the first equation of motion, $v=u+at..........\left(ii\right)$

Substituting (ii) in (i),
$s=\left(\frac{u+(u+at)}{2}\right)t$

$\therefore s=ut+\frac{1}{2}a{t}^2$