Home Repair. You need to extend a 2.50-inch-diameter pipe, but you have only a 1.00-inch-diameter pipe on hand. You make a fitting to connect these pipes end to end. If the water is flowing at 6.00 cm/s in the wide pipe, how fast will it be flowing through the narrow one?

Verified step by step guidance

1. Identify the diameters of the two pipes and convert them into consistent units if necessary. In this case, convert the diameters from inches to centimeters: 2.50 inches is approximately 6.35 cm and 1.00 inch is approximately 2.54 cm.
2. Use the formula for the area of a circle, \(A = \pi r^2\), to calculate the cross-sectional areas of both pipes. Remember to first calculate the radius of each pipe by dividing the diameter by 2.
3. Apply the principle of conservation of mass, which implies that the mass flow rate must be constant throughout the pipe system. This can be expressed as \(A_1 v_1 = A_2 v_2\), where \(A_1\) and \(A_2\) are the cross-sectional areas of the wide and narrow pipes, respectively, and \(v_1\) and \(v_2\) are the flow velocities in these pipes.
4. Substitute the known values into the equation from step 3. You know the flow velocity in the wide pipe (\(v_1\)) and the areas of both pipes (\(A_1\) and \(A_2\)) calculated in step 2.
5. Solve the equation for \(v_2\), the flow velocity in the narrow pipe, to find how fast the water will be flowing through it.

Key Concepts

Continuity Equation

The Continuity Equation states that for an incompressible fluid flowing through a pipe, the product of the cross-sectional area and the fluid velocity must remain constant. This principle can be expressed as A1V1 = A2V2, where A is the area and V is the velocity. It implies that if the diameter of the pipe decreases, the velocity of the fluid must increase to maintain the same flow rate.

Cross-Sectional Area

The cross-sectional area of a pipe is the area of a slice taken perpendicular to the flow direction. For a circular pipe, it can be calculated using the formula A = π(d/2)², where d is the diameter. Understanding the cross-sectional area is crucial for applying the Continuity Equation, as it directly influences the velocity of the fluid in different sections of the pipe.

Fluid Dynamics

Fluid dynamics is the study of the behavior of fluids in motion. It encompasses various principles, including the effects of pressure, velocity, and flow rate. In this context, understanding fluid dynamics helps explain how changes in pipe diameter affect the speed of water flow, which is essential for solving the problem of connecting pipes of different sizes.