Consider a jet impinging normally on a plate. As a result of impact of jet, the plate moves in the direction of jet as shown in.


Let v = velocity of the jet

v₁ = velocity of the plate

a = c/s area of jet

Then relative velocity = (v - v₁) as the plate moving in the same direction as that jet.

Force exerted by jet = mass x change in velocity = ρ a (v – v₁) x (v – v₁) – 0)

F = ρ a (v – v₁)²

Work done = Force x displacement / sec = ρ a (v – v₁)² v₁

Kinetic energy of the jet,

K.E. = ½ ( ρ a v) v² = ½  ρ a v³

Efficiency = Work done / Energy supplied = ρ a (v – v₁)² v₁ / ½ ρ a v³

η  =  2 ( v - v₁)² v₁ / v³

The efficient becomes maximum when (v – v1)2 v1 becomes maximum. For maximum efficiency, differentiate (v – v1)2 w.r.t v1 and equate to zero

(v² – 2 v. v₁ + v₁²)

dη  / dv₁ = v² – 4 v. v₁ + 3v₁² = 0 = (v – 3v₁) (v - v₁) = 0

either v - v₁ = 0, which is not practicable

or v- 3v₁ = 0, v = 3v is the condition for maximum efficiency

η_max  = 2 (3v₁ – v₁)² v₁ / (3v₁) x v₁

η_max  = 8 / 27

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