BGRing,
You have it wrong, the 1/2 x Rho x Vsquared is derived by integrating the rate of change of the momentum. Since the momentum is Rho x V, if you integreate it (using integral calculus) you get the 1/2 x Rho x Vsquared.
It is not that half of anything is lost, it is that the area under the momentum vs V curve is calculated by the formula.