Question

An ideal liquid with density \( \rho \) is poured into a cylindrical vessel with cross section \( A_1\) to a level of height \( h \) from the bottom, which has an opening of cross section \( A_2 \).

Find the time it takes for the liquid to flow out

Answer 1

\[\frac{1}{2}\rho v_1^2+\rho gz=\frac{1}{2}\rho v_2^2\quad,\quad A_1v_1=A_2v_2\]\[v_1^2+2gz=v_2^2\]\[v_1^2+2gz=\left(\frac{A_1}{A_2}\right)^2v_1^2\]\[2gz=\left[\left(\frac{A_1}{A_2}\right)^2-1\right]v_1^2\]\[v_1=\sqrt{\frac{2gz}{(A_1/A_2)^2-1}}\quad,\quad v_1=-\frac{dz}{dt}\]\[-\frac{dz}{dt}=\sqrt{\frac{2gz}{(A_1/A_2)^2-1}}\]\[t=-\sqrt{\frac{(A_1/A_2)^2-1}{2g}}\int\limits_h^0\frac{dz}{\sqrt{z}}\]\[t=-\sqrt{\frac{(A_1/A_2)^2-1}{2g}}\cdot 2\sqrt{z}|_h^0=\sqrt{\frac{2h}{g}\left[\left(\frac{A_1}{A_2}\right)^2-1\right]}\]