§ 30. Torricelli's Theorem & Hydrodynamic Efflux of a Liquid Jet (Grade 7–10)
Efflux speed v = √(2gh) from a vessel orifice and parabolic jet range X = 2√(h·y₀). Liquid emerging from a narrow orifice at depth h below the free surface acquires the exact speed of a body freely falling from height h.
Theory Summary & Scientific Mechanism
In 1643, Evangelista Torricelli discovered that the efflux speed of an ideal liquid jet emerging from a small orifice at depth h below the free surface equals the speed a body would acquire in free fall from height h: v = √(2gh).
Torricelli’s theorem follows directly from Bernoulli’s equation applied along a streamline from the wide free surface (area S₁, elevation H, pressure p₀) to the exit nozzle (area S₂ ≪ S₁, elevation H - h, also at atmospheric pressure p₀): p₀ + ρ g H + (1)/(2)ρ v₁² = p₀ + ρ g (H - h) + (1)/(2)ρ v₂². Since v₁ = v₂(S₂/S₁) ≈ 0 for a wide tank, fluid density ρ cancels out, leaving v₂ = √(2gh).
Launching horizontally from height H - h above the base, the jet falls for time t = √(2(H - h)/g) and strikes at horizontal range L(h) = v t = 2√(h(H - h)). By symmetry, maximum range Lₘax = H occurs when the nozzle is placed exactly at mid-depth (h = H/2).
Everyday Analogy
Poke a small hole near the bottom of a tall water bottle: the water jet shoots out at the exact same speed a droplet would reach if dropped freely from the top water surface! Higher water level means higher pressure and a faster jet.
Interactive Question & Derivation
In a vertical vessel filled with water to total height H, small nozzles are placed at various depths h below the free surface (so nozzle height above the table is y₀ = H - h). At what depth h does the emerging water jet achieve the maximum horizontal range Xₘax on the table?
- A: Right at the very bottom (h → H, y₀ → 0), where hydrostatic pressure and efflux speed v = √(2gh) are greatest
- B: Exactly at mid-height of the liquid column (h = H/2 = y₀), where X = 2√(h(H - h)) reaches its maximum Xₘax = H
- C: Near the top surface (h → 0), because the jet has the longest fall time t = √(2y₀/g)
Study Hint: By Torricelli’s theorem v = √(2gh), efflux speed depends on depth h below the free surface, how does jet speed change as water depth h increases?
Proof & Explanation: By Torricelli's Theorem (derived from Bernoulli's equation), horizontal efflux speed at depth h is v = √(2gh). Falling from nozzle height y₀ = H - h takes time t = √(2y₀ / g). Thus the horizontal range is X = v t = √(2gh)·√(2(H-h)/g) = 2√(h(H - h)). Notice that X is independent of g and Liquid density, is symmetric for h and H - h, and reaches its maximum Xₘax = H at h = H/2!
Key Equations & 3D Simulation Parameters
v = √(2gh), X(h) = 2√(h y₀), Q = A₀√(2gh)
- Water column height H: 2.6 … 5 m (default: 4.4 m)
- Probed nozzle depth h: 0.4 … 4.2 m (default: 2.2 m)
- Nozzle velocity coeff. φ: 0.8 … 1 (default: 0.98)
- Liquid density ρ: 0.8 … 1.8 g/cm³ (default: 1 g/cm³)