Question

Prove that s=ut+1/2 at²

Answer 1

Distance travelled = Area under the line
= ut + ½ (v-u)t
Acceleration (a) = (v-u)/t and so (v-u) = at
Therefore,
Distance travelled (s) = ut + ½ (v-u)t = ut + ½ (at)t = ut + ½ at²
Thus,proved.

Answer 2

acceleration is defined as change in velocity per unit time.
            =>   (v - u) / (t - 0)  = a        =>  v = u + at

Distance traveled is = average velocity * time duration

           $S = \frac{(initial + final velocity)}{2} * duration \\ \\ S = \frac{u + v}{2} * t  \\ \\ S = \frac{1}{2} (u + u + a t ) * t \\ \\ S = \frac{2ut + a t^2}{2} \\ \\ S = u t + \frac{1}{2} a t^2$