(a) A water hose 2.00 cm in diameter is used to fill a...

(a) A water hose $2.00 \mathrm{~cm}$ in diameter is used to fill a 20.0-L bucket. If it takes $1.00$ min to fill the bucket, what is the speed $v$ at which water moves through the hose? $\left(\right.$ Note $\left.: 1 \mathrm{~L}=1000 \mathrm{~cm}^{3} .\right)$ (b) If the hose has a nozzle $1.00 \mathrm{~cm}$ in diameter, find the speed of the water at the nozzle.

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Key Concepts

Volumetric Flow Rate

Volumetric flow rate is the measure of the volume of fluid that passes through a cross-section per unit time. It is calculated as the volume of fluid divided by the time it takes to pass, and it serves as a critical basis for analyzing fluid motion in various applications, including water distribution systems and pipe flows.

Continuity Equation

The continuity equation arises from the principle of mass conservation in fluid dynamics. For an incompressible fluid, it states that the product of the cross-sectional area and the velocity of the fluid remains constant along a streamline. This concept is essential for understanding how fluid speeds change when the area of the passage changes, such as when flowing from a hose to a nozzle.

Cross-Sectional Area Calculation

Calculating the cross-sectional area of a pipe or nozzle, typically using the formula A = ?*(diameter/2)^2, is fundamental in fluid dynamics. This calculation is crucial for converting between volumetric flow rate and fluid velocity, as the area directly influences the speed at which the fluid moves through different sections of a system.

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