Question

Find the height of the water in the long tube above the top when the water stops coming out of the hole.

• A. $-2h_0$
• B. $h_0$
• C. $h_2$
• D. $-h_1$

Answer 1

The correct option is D $-h_1$

Equating the pressure due air and due to water we get
$2p_0=(h_2+h_0)\rho g+p_0$ (since liquids at the same level have the same pressure)
$p_0=h_2\rho g+h_0\rho g$
$h_2\rho g=p_0-h_0\rho g$
$h_2=\frac{p_0}{\rho g}-\frac{h_0\rho g}{\rho g}=\frac{p_0}{\rho g}-h_0$
KE of the water $=$ Pressure energy of the water at that layer
$\frac{1}{2}m{v}^2=m\times \frac{P}{\rho }$
$v^2=\frac{2P}{\rho }=\frac{2}{\rho }[p_0+\rho g(h_1-h_0\left)\right]$
$v={\left[\frac{2}{\rho }\{p_0+\rho g(h_1-h_0\left)\right\}\right]}^{\frac{1}{2}}$

Now we know:
$2P_0+\rho g(h_1-h_0)=p_0+\rho gX$
$\Rightarrow X=\frac{p_0}{\rho g}+(h_1-h_0)=h_2+h_1$
i.e, X is $h_1$ metre below the top or X is $-h_1$ above the top.