Problem $4$: A stationary water tank of diameter $D=1m$ is mounted on wheels and is placed on a nearly frictionless level surface. A smooth hole of diameter $D_0=0\mathrm{.}01m$ near the bottom of the tank allows water to jet horizontally and rearward and the water jet force propels the system forward. The water in the tank is much heavier than the tank$-$and$-$wheel assembly, so only the mass of water remaining in the tank needs to be considered in this problem. By taking point $1$ at the free surface of the tank, and point $2$ at the outlet of the hole, which is also taken to be the reference level $)=(0$ so that the water height above the hole at any time is $z.$ Considering the water height above the hole at any time is $z=3m,$ calculate the discharge rate of water from the tank through the hole. The density of water is $1000k\frac{g}{m}3,g=9\mathrm{.}80655\frac{m}{s^2},=3\mathrm{.}14159,m=V=z(\frac{D^2}{4}).$

Mass Balance: $,{\displaystyle \underset{in}{\overset{?}{}}}{m}^{}={\displaystyle \underset{\text{o}\text{u}\text{t}}{\overset{?}{}}}{m}^{},$;$,m^{}=V^{}=VA$

Bernoulli Equation:

$\frac{P_1}{}+\frac{V_1^2}{2}+g{z}_1=\frac{P_2}{}+\frac{V_2^2}{2}+g{z}_2$

$or$

$\frac{P_1}{g}+\frac{V_1^2}{2g}+z_1=\frac{P_2}{g}+\frac{V_2^2}{2g}+z_2$

$or$

$P_1+\frac{V_1^2}{2}+g{z}_1=P_2+\frac{V_2^2}{2}+g{z}_2$