Question 419: Properties of Matter

Question

Two large, identical water tanks, 1 and 2 , kept on the top of a building of height $H$, are filled with water up to height $h$ in each tank. Both the tanks contain an identical hole of small radius on their sides, close to their bottom. A pipe of the same internal radius as that of the hole is connected to tank 2 , and the pipe ends at the ground level. When the water flows from the tanks 1 and 2 through the holes, the times taken to empty the tanks are $t_1$ and $t_2$, respectively. If $H=\left(\frac{16}{9}\right) h$, then the ratio $t_1 / t_2$ is ___________.

Accepted Answer

$$\begin{aligned}
& A v=\mathrm{av}_1 \
& A\left(-\frac{d y}{d t}ight)=a \sqrt{2 g y} ; d t=\frac{A}{a \sqrt{2 g}} \cdot \frac{-d y}{\sqrt{y}} \
& \int_\limits 0^{t_1} d t=\frac{A}{a \sqrt{2 g}} \int_\limits h^0-\frac{d y}{\sqrt{y}} \
& t_1=\frac{A}{a \sqrt{2 g}} 2 \sqrt{h} ; t_1=\frac{A}{a} \sqrt{\frac{2 h}{g}}
\end{aligned}$$
$$A v^{\prime}=a v_2$$
$$
\begin{aligned}
& A\left(-\frac{d y}{d t}ight)=a \sqrt{2 g(H+y)} \
& d t=-\frac{A}{a \sqrt{2 g}} \frac{d y}{\sqrt{H+y}} \
& \int_\limits0^{t_2} d t=-\frac{A}{a \sqrt{2 g}} \int_\limits{\mathrm{H}}^0 \frac{d y}{\sqrt{H+y}} \
& t_2=\frac{A}{a \sqrt{2 g}}(2)(\sqrt{H+h}-\sqrt{H}) \& H=\frac{16 h}{9} \
& =\frac{A}{a} \sqrt{\frac{2 h}{g}}\left(\frac{5}{3}-\frac{4}{3}ight) \
& t_2=\frac{A}{a} \sqrt{\frac{2 h}{g}}\left(\frac{1}{3}ight) \
& 	ext { ratio } \frac{t_1}{t_2}=3
\end{aligned}$$

Properties of Matter

Integer / Numeric Answer Type