The displaced fluid volume $V$
is the crosssectional area $A$
times the thickness $x$. This
volume remains constant for an incompressible fluid, so
\[
\large{V = A_1 x_1 = A_2 x_2.}
\]

(3)

Using Eq.(3) in Eq.(2) we have
\[
\large{\Delta W = \left( P_1  P_2 \right) V.}
\]

(4)

Since work has been done, there has been a change in the mechanical energy of the
fluid segment. This energy change is found with the help of the next diagram.
The energy change between the initial and final positions is given by
\[
\large{\Delta E = E_2  E_1 =
\left( U_2 + K_2 \right)  \left( U_1 + K_1 \right)}
\]

(5)

where $U$ and
$K$ are the potential and kinetic
energy, respectively.
The potential energy of each fluid segment
$U = mgh$ where
$g$ is the acceleration of gravity,
and $h$ is average fluid height. The
kinetic energy is $K = \frac{1}{2}mv^2$
where $m$ is the fluid mass and
$v$ is the speed of the fluid.
Substituting into Eq. (5), we write
\[
\large{\Delta E = \left(m_2gh_2 + \frac{1}{2}m_2 v_2^2 \right)
 \left(m_1gh_1 + \frac{1}{2}m_1 v_1^2 \right).}
\]

(6)

The workenergy theorem says that the net work done is equal to the change in the
system energy. This can be written as
\[
\large{\Delta W = \Delta E.}
\]

(7)

Substitution of Eq.(4) and Eq.(6) into Eq.(7) yields
\[
\large{\left( P_1  P_2 \right) V =
\left( m_2gh_2 + \frac{1}{2} m_2 v_2^2 \right) 
\left( m_1gh_1 + \frac{1}{2}m_1 v_1^2 \right).}
\]

(8)

Dividing Eq.(8) by the fluid volume,
$V$ gives us
\[
\large{ P_1  P_2 =
\left( \rho g h_2 + \frac{1}{2} \rho v_2^2 \right) 
\left( \rho g h_1 + \frac{1}{2} \rho v_1^2 \right)}
\]

(9)

where
\[
\large{ \rho = \frac{m}{V}}
\]

(10)

is the constant fluid mass density. To complete our derivation,
we reorganize Eq.(9).
\[
\large{ P_1 + \rho g h_1 + \frac{1}{2} \rho v_1^2 =
P_2 + \rho g h_2 + \frac{1}{2} \rho v_2^2. }
\]

(11)

Finally, note that Eq.(11) is true for any two positions. Therefore,
\[
\large{ P + \rho g h + \frac{1}{2} \rho v^2 = \text{constant.} }
\]

(11)

Equation (11) is commonly referred to as Bernoulli's equation. Keep in mind that
this expression was restricted to incompressible fluids and smooth fluid flows.
