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How does Torricelli’s law predict the speed of water leaving a hole in a container?

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differential manometer, Torricellian barometer, and siphon — fluid mechanics Figure 1: Schematic representations of (A) a differential manometer, (B) a Torricellian barometer, and (C) a siphon.

Torricelli’s law states that the speed, v, of a liquid flowing out of an opening in a container is proportional to the square root of the vertical distance, h, between the liquid surface and the center of the opening, and the square root of twice the acceleration due to gravity, g.1

The relationship is expressed by the equation:
$v = \sqrt{2 g h}$

This law implies that the speed of the efflux is equivalent to the velocity a drop of water would attain if it fell freely under the influence of gravity alone through the same vertical distance, h. This result is a direct consequence of the principle of energy conservation, which is fundamental to Bernoulli’s law in the study of fluid mechanics. The speed of the exiting fluid is independent of the direction of the opening, whether it is directed upward, downward, or horizontally.12