Deduce the equations of unifromly accelerated motion in one dimension by following calculus method.

Explanation

By definition
`a = (dv)/(dt)`
`dv = a dt`
Integrating both sides
`int_(v_(0))^(v) dv = int_(0)^(t) a dt`
`= a int_(0)^(t) dt` (a is constant)
`v - v_(0) = at`
`v = v_(0) + at`
Further, `v = (dx)/(dt)` ltbr. `dx = v dt`
Integrating both sides
`int_(x_(0))^(x) dx = int_(0)^(t) v dt`
`= int_(0)^(t) (v_(0) + at) dt`
`x - x_(0) = v_(0) t + (1)/(2) a t^(2)`
`x = x_(0) + v_(0) t + (1)/(2) a t^(2)`
We can write
`a = (dv)/(dt) = (dv)/(dx) (dx)/(dt) = v (dv)/(dx)`
or, `v dv = a dx`
Integrating both sides
`int_(v_(0))^(v) vdv = int_(x_(0))^(x) a dx`
`(v^(2) - v_(0)^(2))/(2) = a (x - x_(0))`
`v^(2) = v_(0)^(2) + 2a (x - x_(0))`
The advantage of this method is that it can be used for motion with non-uniform acceleration also.
Now, we shall use these equations to some important cases.