Answer:
v₁ = √[ 2gh / ((A₁ / A₂)² − 1) ]
Explanation:
Use Bernoulli's equation:
P₁ + ½ ρ v₁² + ρgz₁ = P₂ + ½ ρ v₂² + ρgz₂
Since there's no elevation change between points 1 and 2, z₁ = z₂.
P₁ + ½ ρ v₁² = P₂ + ½ ρ v₂²
Assuming incompressible fluid, the volumetric flow rate is the same at points 1 and 2.
Q₁ = Q₂
v₁ A₁ = v₂ A₂
v₂ = v₁ A₁ / A₂
Substituting:
P₁ + ½ ρ v₁² = P₂ + ½ ρ (v₁ A₁ / A₂)²
P₁ + ½ ρ v₁² = P₂ + ½ ρ v₁² (A₁ / A₂)²
P₁ − P₂ = ½ ρ v₁² (A₁ / A₂)² − ½ ρ v₁²
P₁ − P₂ = ½ ρ v₁² ((A₁ / A₂)² − 1)
v₁² = 2 (P₁ − P₂) / (ρ ((A₁ / A₂)² − 1))
v₁² = 2 (ρgh) / (ρ ((A₁ / A₂)² − 1))
v₁² = 2gh / ((A₁ / A₂)² − 1)
v₁ = √[ 2gh / ((A₁ / A₂)² − 1) ]
