Suppose there is a Horizontal pipe of Length $L$ which pumps water through an end with a speed $v$, there is no friction. A pump is on the opposite side.
So what will the rate at which of kinetic energy will be imparted to the water which is equal to power of the engine according to this answer on the stack exchange
$$\frac 12\rho AV^3 $$ where
$\rho=$ density of the liquid and $A$ stands for the area of cross section of water.
But according to the traditional formula of power which is equal to
$$\text{force}\times\text{velocity}$$
$$\rho AV^3$$ should be the power of engine
As $$ \rho AV^2 $$ denoting Force multiplied with Velocity $v$
Why there is a difference of factor of $\frac 12$? I have researched on the internet and on stack exchange there is a similar question but answers are not very useful as question is closed for some reason.
My take on this, like if we take a column of length $L$ of water it is so when half the column flows out of the pipe, space created is filled by a new column so Half of the engine power goes into accelerating that column of water so engine works double but power transmitted to the water column of length $L$ is half
