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.

v₂v₁  = A₁A₂ 

Where A₁ = 𝜋 r₁² and A₂ = 𝜋 r₂² so,

v₂v₁  = πr₁²πr₂²

v₂ = v₁ r₁²r₂²

v₂ = v₁ (d₁2)²(d₂2)²      [r₁ = d₁/2 and r₂ = d₂/2]

or

v₂ = v₁ d₁²d₂²

v₂ = v₁ (d₁d₂)² ---------------(1)

by putting values 

v₂ = v1 (0.096m)²(0.025m)²

v₂ = 1.3 m s⁻¹ｘ 0.009216m²0.000625m²

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 (d1d₂)²

or

d₂ = d₁√v₁v₂

by putting values 

d₂ = 0.096 m ｘ√1.3ms⁻¹21ms⁻¹

d2 = 0.096 m ｘ √0.0619

d₂ = 0.096 m ｘ 0.2488

d₂ = 0.02388 m

or

d₂ = 2.4 m---------------Ans.


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