## The Bernoulli Equation - Work and Energy

**Introduction: **Work and energy we know that if we drop a ball it accelerates downward with an acceleration g = 9.81m s^{2}. / (neglecting the frictional resistance due to air). We can calculate the speed of the ball after falling a distance h by the Formula v^{2}u ^{2}= 2 as (a = g and s = h). The equation could be applied to a falling droplet of water as the same laws of motion apply

A more general approach to obtaining the parameters of motion (of both solids and fluids) is to apply the principle of conservation of energy. When friction is negligible the sum of kinetic energy and gravitational potential energy is constant

Gravitational potential energy = m g h

(M is the mass, v is the velocity and h is the height above the datum)

To apply this to a falling droplet we have an initial velocity of zero, and it falls through a height of h.

Initial kinetic energy = 0

Initial potential energy = m g h

Final kinetic energy =1/2mv^{2}Final potential energy = 0

We know that: kinetic energy potential energy = constant so

Initial kinetic energy Initial potential energy = Final kinetic energy Final potential energy

Although this is applied to a drop of liquid, a similar method can be applied to a continuous jet of liquid.

We can consider the situation as in the figure above - a continuous jet of water coming from a pipe with velocity u_{1}. One particle of the liquid with mass m travels with the jet and falls from height z_{1} to z_{2}. The velocity also changes from u_{1} to u_{2}. The jet is travelling in air where the pressure is everywhere atmospheric so there is no force due to pressure acting on the fluid. The only force which is acting is that due to gravity. The sum of the kinetic and potential energies remains constant (as we neglect energy losses due to friction) so

As m is constant this becomes

This will give a reasonably accurate result as long as the weight of the jet is large compared to the frictional forces. It is only applicable while the jet is whole - before it breaks up into droplets.