An ideal fluid of density 800kgm^-3, flows smoothly through a bent pipe (as shown in figure) that tapers in cross-sectional area from a to $\frac{a}{2}.$ The pressure difference between the wide and narrow sections of pipe is 4100 Pa. At wider section, the velocity of fluid is $\frac{\sqrt[]{x}}{6}m{s}^{-1}$  for x = ________. (Given g = 10 ms^-2)

$\rho =800kg/m^3$

P₁ – P₂ = 4100 Pa

 Bernoulli's question b/w (1) & (2)

$\frac{P_1}{\rho g}+\frac{v_1^2}{2g}+z_1=\frac{P_2}{\rho g}+\frac{v_2^2}{2g}+z_2$  

⇒  $\frac{P_1-P_2}{\rho g}+\frac{v_1^2}{2g}+1=\frac{v_2^2}{2g}+0$            -(1)

Equation of continuity

A₁v₁ = A₂v₂

v₂ = 2v₁                -(2)

from (1) & (2)

$\Rightarrow \frac{P_1-P_2}{\rho g}+\frac{v_1^2}{2g}+1=\frac{{\left(2v_1\right)}^2}{2g}$  

$\frac{4100}{800*10}+\frac{v_1^2}{2g}+1=\frac{4v_1^2}{2g}$  

$\frac{12100}{8000}=\frac{3v_1^2}{2g}$

$\Rightarrow v_1^2=\frac{2*10}{3}*\frac{12100}{8000}$  

  $=\frac{2}{3}*\frac{121}{8}$  

                 x = 363