Tuesday, January 23, 2018

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Fluid Mechanics - Torricelli’s Equation

The velocity of a liquid stream exiting from a nozzle, pressured solely by a vertical column of that same liquid, is equal to the free-fall velocity of a solid mass dropped from the same height as the top of the liquid column. In both cases, potential energy (in the form of vertical height) converts to kinetic energy (motion):

 

 

This was discovered by Evangelista Torricelli almost 100 years prior to Bernoulli’s more comprehensive formulation. The velocity may be determined by solving for v after setting the potential and kinetic energy formulae equal to each other (since all potential energy at the upper height must translate into kinetic energy at the bottom, assuming no frictional losses):

 

 

Note how mass (m) simply disappears from the equation, neatly canceling on both sides. This means the nozzle velocity depends only on height, not the mass density of the liquid. It also means the velocity of the falling object depends only on height, not the mass of the object.

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