Derive the equation `x = x _(0) + ut + (1)/(2) at ^(2) or s = ut + (1)/(2) at ^(2).`

Explanation

Let `x _(0)` is position of particle at t = 0 and x is position at time t.
Distance travelled in time t = Area under OPQS region
= Area of rectangle OPRS + Area of triangle PQR
`= (PR + RS) + (1)/(2) (QR + PR)`
From PR = t
RS = u
`QR = v - u`
`x -x _(0) = ut + (1)/(2) (v - u 0 . t ( because v - u = at)`
`x-x _(0) = ut + (1)/(2) at^(2).`
`s =` distance covered `=x -x _(0) ut + (1)/(2) at ^(2).`