A fluid is flowing through a horizontal pipe of varying cross-section, with speed $v \mathrm{~ms}^{-1}$ at a point where the pressure is $P$ Pascal. At another point where pressure is
$\frac{P}{2}$ Pascal its speed is $\mathrm{V} \mathrm{ms}^{-1}$. If the density of the fluid is
$\rho \mathrm{kg} \mathrm{m}^{-3}$ and the flow is streamline, then $\mathrm{V}$ is equal to:
Correct Option: , 4
(4) Using Bernoulli's equation
$P_{1}+\frac{1}{2} \rho v_{1}^{2}+\rho g h_{1}=P_{2}+\frac{1}{2} \rho v_{2}^{2}+\rho g h_{2}$
For horizontal pipe, $h_{1}=0$ and $h_{2}=0$ and taking
$P_{1}=P, P_{2}=\frac{P}{2}$, we get
$\Rightarrow P+\frac{1}{2} \rho v^{2}=\frac{P}{2}+\frac{1}{2} \rho V^{2}$
$\Rightarrow \frac{P}{2}+\frac{1}{2} \rho v^{2}=\frac{1}{2} \rho V^{2} \Rightarrow V=\sqrt{v^{2}+\frac{P}{\rho}}$
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