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Ch. 13 - Fluids
Giancoli Douglas - Physics for Scientists and Engineers 5th edition
Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook
All textbooksGiancoli Douglas 5th editionCh. 13 - FluidsProblem 83
Chapter 13, Problem 83

A pump supplies water to a 1.59-cm inner diameter hose that tapers down to a 0.953-cm-diameter nozzle. The nozzle is aimed so water comes out at a 45° angle and lands 3.0 m away. The nozzle is 0.60 m above ground level, and the pump output is essentially at ground level. What pressure is supplied by the pump?

Verified step by step guidance
1
Determine the velocity of water exiting the nozzle using projectile motion principles. The horizontal range (3.0 m), the angle of projection (45°), and the height difference (0.60 m) are given. Use the kinematic equations to find the time of flight and horizontal velocity. Start with the vertical motion equation: y = v_y t - \(\frac{1}{2}\) g t^2, where y = -0.60 \(\text{ m}\), v_y = v \(\sin\)(45°), and g = 9.8 \(\text{ m/s}\)^2. Solve for time t.
Using the time of flight from step 1, calculate the horizontal velocity v_x using the horizontal motion equation: x = v_x t, where x = 3.0 \(\text{ m}\) and v_x = v \(\cos\)(45°). Combine this with the result from step 1 to find the total velocity v of water exiting the nozzle.
Apply the principle of conservation of mass to relate the velocity of water in the hose to the velocity of water exiting the nozzle. The flow rate is constant, so A_1 v_1 = A_2 v_2, where A_1 and A_2 are the cross-sectional areas of the hose and nozzle, respectively, and v_1 and v_2 are the corresponding velocities. Use the diameters of the hose and nozzle to calculate A_1 and A_2, and solve for v_1.
Use Bernoulli's equation to relate the pressure at the pump to the pressure at the nozzle. Bernoulli's equation is: 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 P_2 is atmospheric pressure, h_1 = 0, and h_2 = 0.60 \(\text{ m}\). Substitute the known values and solve for P_1, the pressure supplied by the pump.
Combine all the results from the previous steps to express the pressure supplied by the pump, P_1, in terms of the given quantities and constants. Ensure all units are consistent, and simplify the expression as much as possible.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Bernoulli's Principle

Bernoulli's Principle states that in a flowing fluid, an increase in the fluid's speed occurs simultaneously with a decrease in pressure or potential energy. This principle is crucial for understanding how the pressure changes as water flows through the varying diameters of the hose and nozzle, affecting the velocity and pressure of the water exiting the nozzle.
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Continuity Equation

The Continuity Equation is based on the conservation of mass, stating that the mass flow rate must remain constant from one cross-section of a pipe to another. In this scenario, as water flows from the wider hose to the narrower nozzle, its velocity increases, which is essential for calculating the speed of water exiting the nozzle and subsequently the pressure required from the pump.
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Projectile Motion

Projectile Motion describes the motion of an object that is launched into the air and is subject to gravitational acceleration. In this problem, understanding the trajectory of the water as it exits the nozzle at a 45° angle and lands 3.0 m away is vital for determining the initial velocity of the water, which is necessary for calculating the pressure supplied by the pump.
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Related Practice
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