FYKOS Year 26 Series 6 Problem 4 — filling a tank · FYKOS 26 #26.6.4

Physics · mechanics

Problem

Imagine a large tank containg tea with a little opening at its bottom so that one can pour it into a glass. When open, the speed of the flow of tea from the tank is v0v_{0}. How will this speed change if, while pouring a glass of tea, someone is filling the tank by pouring water into it from its top? Assume that the diameter of the tank is DD, the diameter of the flow of tea into the tank is dd, and that of the flow of tea out of the tank is much smaller than DD. The tea level is height HH above the lower opening, and the tank is being filled by pouring a water into it from height hh above the tea level. You are free to neglect all friction.
Intuitive explanation

This problem hinges on Torricelli's law and understanding how the tea level changes when inflow and outflow compete. The initial speed v0=2gHv_0 = \sqrt{2gH} depends on the height of tea in the tank, so you need to find how that height evolves when water pours in from above. The key insight is to use continuity of mass: the rate at which the tea level rises (or falls) depends on the net flow—inflow minus outflow—relative to the tank's cross-sectional area. Once you track how HH changes with time, you can determine the new outflow speed.

Intuitive explanation

This problem hinges on Torricelli's law and understanding how the tea level changes when inflow and outflow compete. The initial speed v0=2gHv_0 = \sqrt{2gH} depends on the height of tea in the tank, so you need to find how that height evolves when water pours in from above. The key insight is to use continuity of mass: the rate at which the tea level rises (or falls) depends on the net flow—inflow minus outflow—relative to the tank's cross-sectional area. Once you track how HH changes with time, you can determine the new outflow speed.

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