Torricelli's Law and Draining Tanks

A hole near the bottom of a bucket squirts harder because there's more water piled on top of it, pushing down. That extra depth means extra pressure, and extra pressure means extra speed when the water finally escapes. Bernoulli's equation makes this exact. Compare the open top of the tank to the hole: both are open to the air, so both sit at atmospheric pressure, and the pressure terms cancel. The surface drops slowly if the tank is wide, so its speed is nearly zero. What's left is gy_top + 0 = gy_hole + v²/2, which rearranges to v = √2gh, where h is the depth of the hole below the surface. That's just the free-fall speed formula. Physically, each bit of fluid converts its lost height into speed the same way a dropped rock does, even though it's moving sideways out a hole instead of straight down.

Once the water leaves the hole, Bernoulli's job is done. The stream is now a projectile: it exits horizontally with speed v = √2gh, then falls under gravity like anything else launched sideways off a ledge. The height of the hole above the ground sets the fall time, and that time times v gives the landing distance. This is why the topic sits right after pressure-with-depth and right before you'd treat the squirting water as ordinary projectile motion: Torricelli's law supplies the launch speed, and kinematics takes it from there.

Putting a hole too close to the surface gives high fall time but low speed; too close to the ground gives high speed but almost no fall time. Somewhere in between is a sweet spot, and that tradeoff is what range-maximizing problems are testing.