Question

Selena uses a garden hose to fill a bucket of water. The water enters the hose through a faucet with a 6.0-cm diameter. The speed of the water at the faucet is 5 m/s. If the faucet and the nozzle are at the same height, and the water leaves the nozzle with a speed of 20 m/s, what is the diameter of the nozzle?
I keep coming up with 4 cm. Please someone confirm my answer. If incorrect please explain why.

Answer 1

Answer:

D₂ = 2.738 cm

Explanation:

Continuity equation

The continuity equation is nothing more than a particular case of the principle of conservation of mass. It is based on the flow rate (Q) of the fluid must remain constant throughout the entire pipeline.

Since the flow rate is the product of the surface of a section of the duct because of the speed with which the fluid flows, we will have to comply with two points of the same pipeline:  

Q = v*A  : Flow Equation

where:

Q = Flow in (m³/s)

A is the surface of the cross sections of points 1 and 2 of the duct.

v is the flow velocity at points 1 and 2 of the pipe.

It can be concluded that since the flow rate must be kept constant throughout the entire duct, when the section decreases, the flow rate increases in the same proportion and vice versa.

Data

D₁= 6.0 cm : faucet  diameter

v₁ = 5 m/s :  speed of fluid in the  faucet

v₂ = 20 m/s : speed of fluid in the  nozzle

Area calculation

A = (π*D²)/4

A₁ = (π*D₁²)/4

A₂ = (π*D₂²)/4

Continuity equation  

Q₁ = Q₂

v₁A₁ = v₂A₂

v₁(π*D₁²)/4 = v₂(π*D₂²)/4 : We divide by (π/4) both sides of the equation

v₁ (D₁)² = v₂(D₂)²

We replace data

6 *(5)² = 20*(D₂)²

150 = 20*(D₂)²

(150 /20) = (D₂)²

7.5 = (D₂)²

$D_{2} = \sqrt{7.5}$

D₂ = 2.738 cm :  nozzle diameter