Let's break down the problem step by step. ### **Given Data** - **Pressure at top of container** (\(P_{top}\)): \(2 \, \text{atm}\) - **Height of fluid column** (\(h_0\)): \(10\, \text{m}\) - **Vertical distance to outlet** (\(d\)): \(5\, \text{m}\) - **Radius of tube** (\(r\)): \(2\, \text{cm} = 0.02\, \text{m}\) - **Fluid**: Water (\(\rho = 1000\, \text{kg/m}^3\)) - **Acceleration due to gravity** (\(g\)): \(9.81\, \text{m/s}^2\) - **Atmospheric Pressure** (\(P_{atm}\)): \(1\, \text{atm} = 101325\, \text{Pa}\) --- ### **Step 1: Apply Bernoulli’s Equation** Apply Bernoulli's equation between the water surface (point 1) and the outlet (point 2): \[ P_1 + \frac{1}{2}\rho v_1^2 + \rho g h_1 = P_2 + \frac{1}{2}\rho v_2^2 + \rho g h_2 \] Assume the water surface velocity (\(v_1\)) is negligible (\(v_1 \approx 0\)). Let: - \(P_1 = 2\, \text{atm} = 2 \times 101325 = 202650\, \text{Pa}\) - \(P_2 = 1\, \text{atm} = 101325\, \text{Pa}\) (pressure at outlet open to atmosphere) - \(h_1 = h_0 = 10\, \text{m}\) - \(h_2 = d = 5\, \text{m}\) - \(v_2 = v\) (velocity at outlet) Plug in values: \[ P_1 + \rho g h_1 = P_2 + \frac{1}{2}\rho v^2 + \rho g h_2 \] \[ 202650 + 1000 \times 9.81 \times 10 = 101325 + \frac{1}{2} \times 1000 \times v^2 + 1000 \times 9.81 \times 5 \] Calculate each term: - \(1000 \times 9.81 \times 10 = 98100\) - \(1000 \times 9.81 \times 5 = 49050\) So: \[ 202650 + 98100 = 101325 + 49050 + 500 v^2 \] \[ 300750 = 150375 + 500 v^2 \] \[ 300750 - 150375 = 500 v^2 \] \[ 150375 = 500 v^2 \] \[ v^2 = \frac{150375}{500} = 300.75 \] \[ v = \sqrt{300.75} \approx 17.34\, \text{m/s} \] --- ### **Step 2: Calculate Flow Rate (\(Q\))** \[ Q = \text{Area} \times \text{Velocity} \] - Area of tube, \(A = \pi r^2 = \pi (0.02)^2 = \pi \times 0.0004 = 0.001257\, \text{m}^2\) - Flow rate: \(Q = 0.001257 \times 17.34 = 0.0218\, \text{m}^3/\text{s}\) --- ### **Final Answers** - **Velocity at outlet (\(v\)):** \(\boxed{17.34\, \text{m/s}}\) - **Flow rate (\(Q\)):** \(\boxed{0.0218\, \text{m}^3/\text{s}}\) --- **Summary of Solution Steps:** 1. **Apply Bernoulli's Equation** to find the velocity at the outlet. 2. **Calculate Flow Rate** using the tube's area and the outlet velocity. Let me know if you’d like any further clarification!