Question

derivation of s=ut+1/2at​

Answer 1

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

Consider a body moving with initial velocity u m/s due to uniform acceleration a m/s² body attains final velocity v m/s in t seconds.

Displacement covered in s metres.

We know that , Average velocity = Displacement/Time

Average velocity = s/t -----(1)

Since body is in uniform acceleration. Average velocity = (Initial velocity + Final velocity)/2

Therefore Average velocity = (u + v)/2 ----(2)

From (1) and (2)

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

But v = u + at ,So, substitute the value.

$\frac{s}{t} = \frac{u + (u + at)}{2}$

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

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

$s = t( \frac{2u + at}{2} )$

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

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

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