State and prove Bernoulli's theorem.

Let us consider the ideal liquid of density ρ flowing through the pipe LM of varying cross-section.
Let P₁ and P₂ be the pressures at ends L and M and A₁ and A₂ be the areas of cross-sections at ends L and M respectively.
Let the liquid enter with velocity V₁ and leave with velocity v₂.
Let A₁ > A₂.
Now, by equation of continuity,
                      
Since, A₁ > A₂
Therefore, 
v₂ > v₁ and P₁ > P₂ 
Let, m be the mass of liquid enetring at end L in time t. 
The liquid will cover a distance = v₁t 
Therefore, the work done by pressure on the liquid at end L in time t is, 
W₁ = Force  
      = P₁A₁v₁t           ... (1) 
Since same mass m leaves the pipe at end M in same time t, in which liquid will cover the distance given by v₂t.
Therefore, work done by liquid against the force due to pressure P₁ is given by, 
 
        W₂ = P₂A₂v₂t        ... (2) 
Net ork done by pressure on the liquid in time t is, 
W = W₁ - W₂ 
    =P₁A₁v₁t  - P₂A₂v₂t        ... (3) 
This work done on liquid by pressure increases it's kinetic energy and potential energy. 
Increase in K.E of liquid is, 

That is,