Water flowing through a garden hose of diameter 2.74 cm...

Water flowing through a garden hose of diameter $2.74 \mathrm{~cm}$ fills a 25-L bucket in 1.50 min. (a) What is the speed of the water leaving the end of the hose? (b) A nozzle is now attached to the end of the hose. If the nozzle diameter is one-third the diameter of the hose, what is the speed of the water leaving the nozzle?

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Key Concepts

Calculating Cross-sectional Area

For a circular pipe or hose, the cross-sectional area is calculated using the formula A = ?r², where r is the radius (half of the diameter). This area is critical for connecting the physical dimensions of the hose to the speed and flow rate of the fluid.

Nozzle Effect

When a nozzle is attached to a hose, it reduces the cross-sectional area available for the flow. According to the continuity equation, this reduction forces the fluid to increase its speed to maintain the same volumetric flow rate, illustrating how constrictions in a flow path can accelerate fluid velocity.

Volumetric Flow Rate

Volumetric flow rate is the volume of fluid passing through a cross-sectional area per unit time. It is expressed as the product of the fluid’s velocity and the cross-sectional area, providing a direct relationship that allows the computation of one variable if the others are known.

Continuity Equation

This principle in fluid dynamics states that for an incompressible fluid, the product of the cross-sectional area and the speed of the fluid remains constant along a streamline. It explains why a decrease in cross-sectional area (as when using a nozzle) must result in an increase in fluid speed in order to maintain the same flow rate.

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