Discover the power of flowing water
Sluice: Q = Cd b w √(2 g hc) (Torricelli), hc = pond level above the centre of the gate opening w. Water power available: P = ρ g Q H.
Undershot: only the jet's kinetic energy acts: P = ρ Q (v − u) u, best at u = v/2 and at most 50 % of the jet's energy. Overshot / breastshot: filled buckets also carry weight down the wheel: torque ≈ ρ g z Q / ω, limited by how much the buckets can hold per turn (Q ≤ Vb ω/2π) and spilling at high rim speed.
Why head matters: a wheel fed too far below the pond surface wastes the extra head as a fast jet (a breastshot wheel at 3.6 m head); a wheel fed too high leaves no water. Match head and wheel: overshot ≈ 60 %, breastshot ≈ 40 %, undershot ≈ 25 % here (Smeaton measured 63 % / ~50 % / 22 % in the 1750s).
Dynamics: J dω/dt = τwater − τload; pond: A dH/dt = Qin − Q when not held. Flour rate = shaft power ÷ 0.06 kWh/kg.
Simplifications: the numbers are a teaching model with fitted loss factors (illustrative, not a design); no ice, leaks, gearing losses or stone wear; tail-water backing up is ignored.