Physics

Physics

Problem

A vertical cylindrical tank has a constant cross-sectional area of 0.50 m20.50\,\text{m}^2. A small orifice of area 1.0 cm21.0\,\text{cm}^2 is made at its base. When the depth of water above the orifice is 1.6 m1.6\,\text{m}, what is the speed (magnitude) at which the water level is falling? a) 2.2 mm s−12.2\,\text{mm}\,\text{s}^{-1} b) 0.67 mm s−10.67\,\text{mm}\,\text{s}^{-1} c) 1.1 mm s−11.1\,\text{mm}\,\text{s}^{-1} d) 6.3 mm s−16.3\,\text{mm}\,\text{s}^{-1}
Intuitive explanation

This problem combines fluid dynamics with the continuity equation—the key is recognizing that water leaving through the orifice must equal the rate at which the tank's water level drops. Use Torricelli's law to find the exit speed through the small hole, then apply conservation of mass: the volume flowing out per unit time equals the rate at which the tank's surface area times height decreases. The answer reveals how a tiny orifice can drain a large tank surprisingly fast.

Intuitive explanation

This problem combines fluid dynamics with the continuity equation—the key is recognizing that water leaving through the orifice must equal the rate at which the tank's water level drops. Use Torricelli's law to find the exit speed through the small hole, then apply conservation of mass: the volume flowing out per unit time equals the rate at which the tank's surface area times height decreases. The answer reveals how a tiny orifice can drain a large tank surprisingly fast.

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