Derive the formula : $s=ut+\frac{ 1 }{ 2 }at^{ 2 }$, where the symbols have usual meanings.

Let a moving body has the initial velocity $u$ and within the time $t$ its velocity becomes $v$ with the acceleration $a$ and let $s$ be the distance covered by the body.

As known acceleration $a=\frac{change\ in\ velocity}{time}$

Or $a=\frac{v-u}{t}$

Or $at=v-u$

Or $v=u+at$

We know the average velocity $=\frac{u+v}{2}$

Distance $s=average\ velocity\times time$

Or $s=\frac{u+v}{2}\times t$

Or $s=\frac{u+u+at}{2}\times t$

Or $s=\frac{2u+at}{2}\times t$

Or $s=\frac{2u}{2}\times t+\frac{1}{2}\times at\times t$

Or $s=ut+\frac{1}{2}at^2$

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Updated on: 10-Oct-2022

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