Obtain equations of motion for constant acceleration using method of calculus.

Explanation

By definition `a=(dupsilon)/(dt)`
`dupsilon=a" "dt`
Integrating both sides
`int_(upsilon_(0))^(upsilon)dupsilon=int_(0)^(t)a" "dt`
`" "=aint_(0)^(t)dt" (a is constant)"`
`upsilon-upsilon_(0)=at`
`" "upsilon=upsilon_(0)+at`
Further, `upsilon=(dx)/(dt)`
`" "dx=upsilon" "dt`
Integrating both sides
`int_(x_(0))^(x)dx=int_(0)^(t)upsilondt`
`" "=int_(0)^(t)(upsilon_(0)+at)dt`
`x-x_(0)=upsilon_(0)t+1/2a" "t^(2)`
`x=x_(0)+upsilon_(0)t+1/2a" "t^(2)`
We can write
`a=(dupsilon)/(dt)=(dupsilon)/(dx)(dx)/(dt)=upsilon(dupsilon)/(dx)`
Or, `upsilondupsilon=a" "dx`
Integrating both sides,
`int_(upsilon_(0))^(upsilon) upsilondv=int_(x_(0))^(x)adx`
`(upsilon^(2)-upsilon_(0)^(2))/2=a(x-x_(0))`
`upsilon^(2)=upsilon_(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.