Derive an expression for power and velocity.

Explanation

The work done by a force `vecF` for a displacement `dvecr` is
`W=intvecF.dvecr` …(1)
Left hand side of the equation (1) can be written as
`W=intdW=int(dW)/(dt)dt` …(2)
Since, velocity is `vecv=(dvecr)/(dt),dvecr=vecvdt`. Right hand side of the equation (1) can be written as `intvecF.dvecr=int(vecF.(dvecr)/(dt))dt=int(vecF.vecv)dt[vecv=(dvecr)/(dt)]` ....(3)
Subsitituting eqution (2) and eqution(3) in equation (1), we get
`int(dW)/(dt)dt=int(vecf.vecv)dt`
`int((dW)/(dt)-vecF.vecv)dt=0`
This relation is true for any arbitrary value of dt. This implies that the term within the bracket must be equal to zero, i.e.,
`(dW)/(dt)-vecF.vecv=0 "or" (dW)/(dt)=vecF.vecv=p`
Hence power `p=vecF.vecv`