Torricelli's experiment and conservation of momentum

Torricelli's law states that the speed of a fluid as it flows out of an orifice of a large reservoir open to the atmosphere is equal to 2gh, where g is the acceleration of gravity and h is the distance between the free surface of the fluid in the reservoir and the orifice. Obtaining the flow rate from this speed is not at all trivial, despite how simple the situation seems, as the fluid jet issued at the orifice does not have a constant cross section and the motion of the fluid near the orifice is not really known. Here, we use Torricelli's experiment as a practical way to illustrate how to properly apply the momentum balance equation to solve hydrodynamic problems. We compare the horizontal component of the force exerted by the container on the fluid using both momentum conservation and the integral of the stress tensor; this results in a contradiction that we use to review, after considering simple experimental results, the assumptions made during the calculations, to finally resolve the discrepancy and rationalize the intricacies of this every-day situation.