• word_count: 180

Torricelli's theorem states that the speed of liquid efflux from a small orifice at depth $h$ below the free surface is $v = \sqrt{2gh}$. Derivation applies Bernoulli between the surface (large area, $v \approx 0$, $P = P_0$) and the orifice ($P = P_0$).

After exit, the stream is a projectile. For a hole at height $y$ above the ground with water surface at height $H$: time of flight $t = \sqrt{2y/g}$, range $R = 2\sqrt{(H-y) \cdot y}$. Maximum range at $y = H/2$ gives $R_{\max} = H$.

Tank draining time involves integration because $h$ decreases as the tank empties. For a tank of cross-section $A$ with a small orifice of area $a$ at the bottom: $t = (A/a)\sqrt{2H/g}$. The key insight is that the tank takes longer to drain the second half than the first because lower head means slower efflux. The ratio of times: first half takes about 29% of total time, second half takes 71%.