Water travels through a 9.6 cm diameter fire hose with a speed of 1.3 m/s. At the end of the hose, the water flows out through a nozzle whose diameter is 2.5 cm. (a) What is the speed of the water coming out of the nozzle? (b) What diameter nozzle is required to give a water speed of 21 m/s? ((a)19 m/s, (b) 2.4 cm) 


Given:

Diameter of fire hose = d₁ = 9.6 cm = 0.096 m
Speed of water at the fire hose = v₁ = 1.3 m s⁻¹
Diameter of nozzle = d₂ = 2.5 cm = 0.025 m

To Find:

(a) Speed of water at nozzle = v₂ = ?
(b) Diameter of nozzle = d₂ = ? when the water speed = v₂ = 21 m s⁻¹

Solution:

(a) Speed of water at nozzle = v₂ = ?


Using the equation of continuity i.e.

`\frac {v₂}{v₁}`  = `\frac {A₁}{A₂}` 

Where A₁ = 𝜋 r₁² and A₂ = 𝜋 r₂² so,

`\frac {v₂}{v₁}`  = `\frac { π r₁²}{ π r₂²}`

v₂ = v₁ `\frac {r₁²}{r₂²}`

v₂ = v₁ `\frac {(d₁/2)²}{(d₂/2)²}`      [r₁ = d₁/2 and r₂ = d₂/2]

or

v₂ = v₁ `\frac {d₁²}{d₂²}`

v₂ = v₁ (`\frac {d₁}{d₂}`)² ---------------(1)

by putting values 

v₂ = v1 `\frac {(0.096 m)²}{(0.025 m)²}`

v₂ = 1.3 m s⁻¹ｘ `\frac {0.009216 m²}{0.000625 m²}`

v₂ = 1.3 m s⁻¹ｘ 14.7456

v₂ = 19.16928 m s⁻¹

or

v₂ = 19 m s⁻¹ ----------Ans.


(b) Diameter of nozzle = d₂ = ? when the water speed = v₂ = 21 m s⁻¹


Using the equation of  continuity

v₂ = v₁ (`\frac {d1}{d₂}`)²

or

d₂ = d₁`\sqrt frac {v₁}{v₂}`

by putting values 

d₂ = 0.096 m ｘ`\sqrt frac {1.3 m s⁻¹}{21 m s⁻¹}`

d2 = 0.096 m ｘ `\sqrt {0.0619}`

d₂ = 0.096 m ｘ 0.2488

d₂ = 0.02388 m

or

d₂ = 2.4 m---------------Ans.


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