A garden hose attached with a nozzle is used to fill 10-gal bucket. The inner diameter of the hose is 2cm and it reduces to 0.8 cm at the nozzle exit

To determine how quickly the 10-gallon bucket can be filled using a garden hose with a nozzle, we need to consider a few key factors: the flow rate of the water through the hose, the cross-sectional areas of the hose and nozzle, and the relationship between these areas and the speed of the water. Let's break this down step by step.

Understanding Flow Rate

The flow rate of water through a hose can be influenced by the diameter of the hose and the nozzle. The flow rate (Q) is defined as the volume of water passing through a cross-section per unit time, typically measured in liters per minute (L/min) or gallons per minute (GPM).

Cross-Sectional Areas

First, we need to calculate the cross-sectional areas of both the hose and the nozzle. The formula for the area (A) of a circle is:

A = π * (d/2)²

where d is the diameter of the circle.

• For the hose: The inner diameter is 2 cm.
• For the nozzle: The diameter reduces to 0.8 cm.

Now, let's calculate these areas:

• Hose area: A_hose = π * (2 cm / 2)² = π * (1 cm)² ≈ 3.14 cm²
• Nozzle area: A_nozzle = π * (0.8 cm / 2)² = π * (0.4 cm)² ≈ 0.50 cm²

Applying the Continuity Equation

The principle of conservation of mass tells us that the flow rate must remain constant throughout the hose and nozzle. This is known as the continuity equation, which states:

A_hose * v_hose = A_nozzle * v_nozzle

where v_hose and v_nozzle are the velocities of the water in the hose and nozzle, respectively.

Calculating Velocities

We can rearrange the equation to find the velocity at the nozzle:

v_nozzle = (A_hose / A_nozzle) * v_hose

Assuming we know the velocity of water in the hose (let's say it's 1 m/s for this example), we can calculate the velocity at the nozzle:

• v_nozzle = (3.14 cm² / 0.50 cm²) * 1 m/s ≈ 6.28 m/s

Calculating Flow Rate

Now that we have the velocity at the nozzle, we can calculate the flow rate:

Q = A_nozzle * v_nozzle

Converting the area from cm² to m² (1 cm² = 0.0001 m²):

• A_nozzle = 0.50 cm² = 0.00005 m²

Now, substituting the values:

Q = 0.00005 m² * 6.28 m/s ≈ 0.000314 m³/s

Converting to Gallons

To convert cubic meters per second to gallons per minute, we use the conversion factor (1 m³ = 264.172 gallons):

Q ≈ 0.000314 m³/s * 264.172 gallons/m³ * 60 s/min ≈ 5.0 GPM

Time to Fill the Bucket

Finally, to find out how long it will take to fill a 10-gallon bucket:

Time = Volume / Flow Rate

Time = 10 gallons / 5.0 GPM = 2 minutes

In summary, with a garden hose that has a nozzle reducing the diameter from 2 cm to 0.8 cm, it would take approximately 2 minutes to fill a 10-gallon bucket, assuming the water flows at a velocity of 1 m/s in the hose. This example illustrates how fluid dynamics principles can be applied to everyday situations like filling a bucket with a garden hose.