Problem 9.71 Challenge

พิจารณาให้ปลายท่อเป็นจุด \(1\) และปากถังเป็นจุด \(2\)

จาก\(P_{1}+\frac{1}{2}ρv_{1}^{2}+ρgy_{1}=P_{2}+\frac{1}{2}ρv_{2}^{2}+ρgy_{2}\)

\(P_{0}+\frac{1}{2}ρv_{1}^{2}+ρg(0)=P_{0}+\frac{1}{2}ρv_{2}^{2}+ρg(H+h)\)

\(ρv_{1}^{2}=\frac{1}{2}\frac{1}{2}ρv_{2}^{2}+ρg(H+h)\)

\(ρv_{1}^{2}=\frac{1}{2}\frac{1}{2}ρv_{2}^{2}+ρgH; H >> h\)

\(v_{1}^{2} = v_{2}^{2} + 2gH\cdots \cdots (1)\)

จาก \(Av_{2} = av_{1}\)

\(v_{2} =\frac{av_{1}}{A}\)

\(v_{2} \approx 0; a << A \cdots \cdots (2)\)

แทน \((2)\) ใน \((1)\) จะได้ \(v_{1}^{2} = 2gH\)

\(v_{1} =\sqrt{2gH}\)

จาก \(\frac{\Delta V}{\Delta t}= Av\)

\(\frac{Ah}{\Delta t}= av_{1}\)

\(\Delta t =\frac{Ah}{av_{1}}\)

\(\Delta t =\frac{Ah}{a2gH}\;\blacksquare\)