# Water is filled in a beaker upto a height of 3 m: Bernoulli's Equation

Problem: Water is filled in a beaker upto a height of 3 m. An orifice (hole) is at a height of 52.5 cm from the bottom of beaker (see figure). The ratio of cross-sectional area of the orifice and the beaker is 0.1. The square of the speed of the liquid coming out from the orifice is, (Take g = 10 m/s2)  (IIT JEE 2005)

1. 50 (m/s)2
2. 50.5 (m/s)2
3. 51 (m/s)2
4. 52 (m/s)2

Answer: The answer is A i.e., the square of the speed of the liquid coming out from the orfice is 50 (m/s)2.

Solution: Let $A_1$ be the cross-sectional area of the beaker and $v_1$ be the flow velocity at a height $h_1={3}\;\mathrm{m}$. Let $A_2$ be the cross-sectional area of the orifice and $v_2$ be the flow velocity at the orifice located at a height $h_2={0.525}\;\mathrm{m}$. Using continuity equation on an imaginary streamline from top of the beaker to the orifice, \begin{align} A_1v_1=A_2v_2, \end{align} we get, \begin{align} \label{lkb:eqn:1} v_1=(A_2/A_1)v_2. \end{align} Bernoulli's equation, \begin{align} P_0+\rho g h_1+\frac{1}{2}\rho v_1^2=P_0+\rho g h_2+\frac{1}{2}\rho v_2^2, \end{align} gives, \begin{align} \label{lkb:eqn:2} v_2^2=v_1^2+2g(h_1-h_2). \end{align} Eliminate $v_1$ from above equations to get, \begin{align} v_2^2&=\frac{2g(h_1-h_2)}{1-({A_2}/{A_1})^2}\\ &=\frac{2(10)(3-0.525)}{1-0.01}\\ &={50}\;\mathrm{m^2/s^2}.\nonumber \end{align}