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

Giancoli Douglas 5th edition

Ch. 13 - Fluids

Problem 93

Chapter 13, Problem 93

You are watering your lawn with a hose when you put your finger over the hose opening to increase the distance the water reaches. If you are holding the hose horizontally, and the distance the water reaches increases by a factor of 4, what fraction of the hose opening did you block?

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Understand the relationship between the range of the water stream and the velocity of the water exiting the hose. The horizontal range of a projectile (in this case, the water) is proportional to the square of its initial velocity: \( R \propto v^2 \). Since the range increases by a factor of 4, the velocity must increase by a factor of \( \sqrt{4} = 2 \).

Recognize that the velocity of the water exiting the hose is related to the pressure and the cross-sectional area of the hose opening. According to the principle of conservation of mass (continuity equation), the flow rate \( Q \) is constant: \( Q = A v \), where \( A \) is the cross-sectional area of the opening and \( v \) is the velocity of the water.

Since the velocity increases by a factor of 2, the cross-sectional area of the opening must decrease to maintain the same flow rate. Specifically, \( A_{new} = \frac{A_{original}}{2} \), because \( v_{new} = 2v_{original} \).

The fraction of the hose opening that was blocked can be determined by comparing the new area to the original area. The blocked fraction is given by \( 1 - \frac{A_{new}}{A_{original}} \). Substituting \( A_{new} = \frac{A_{original}}{2} \), the blocked fraction becomes \( 1 - \frac{1}{2} = \frac{1}{2} \).

Conclude that you blocked half of the hose opening to achieve the increased range of the water stream.

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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 helps explain how blocking part of the hose opening increases the velocity of the water exiting the hose, allowing it to reach a greater distance.

Continuity Equation

The Continuity Equation in fluid dynamics asserts that the mass flow rate must remain constant from one cross-section of a pipe to another. This means that if the cross-sectional area of the hose is reduced by blocking it, the speed of the water must increase to maintain the same flow rate, which is crucial for understanding how the water's reach changes.

Projectile Motion

Projectile Motion refers to the motion of an object that is thrown or projected into the air, influenced only by gravity and its initial velocity. In this scenario, the water behaves like a projectile after leaving the hose, and its horizontal distance is affected by the initial speed, which is increased by blocking the hose opening.

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Related Practice

Textbook Question

A drinking fountain shoots water about 12 cm up in the air from a nozzle of diameter 0.60 cm (Fig. 13–67). The pump at the base of the unit (1.1 m below the nozzle) pushes water into a 1.2-cm-diameter supply pipe that goes up to the nozzle. What gauge pressure does the pump have to provide? Ignore the viscosity; your answer will therefore be an underestimate.

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