Torricelli's Theorem in AP Physics 2

Torricelli's Theorem states that fluid exits an opening in a container at speed v = √(2gh), where h is the height of the fluid surface above the hole. In AP Physics 2 it is a special case of Bernoulli's equation, applied when the container is open to the atmosphere and drains slowly.

What is Torricelli's Theorem?

Torricelli's Theorem gives you the speed of a fluid streaming out of a hole in a container: v = √(2gh), where h is the vertical distance from the fluid's surface down to the opening. Notice what's not in that equation. The density of the fluid doesn't matter, the size of the hole doesn't matter, and the shape of the tank doesn't matter. Only the height of fluid above the hole does.

Here's the intuition that makes it click. √(2gh) is exactly the speed an object reaches after free-falling a height h. Torricelli's Theorem is basically saying the water leaves the hole as fast as if each drop had been dropped from the surface. That's not a coincidence; it's energy conservation. The theorem comes straight from Bernoulli's equation when you assume the fluid is incompressible, the top surface and the hole are both at atmospheric pressure, and the tank is wide enough that the surface barely moves (so its velocity is approximately zero by the continuity equation).

Why Torricelli's Theorem matters in AP® Physics 2

Torricelli's Theorem lives in Topic 1.7, Conservation of Mass Flow Rate in Fluids, inside Unit 1 of AP Physics 2. It matters because it ties together the two big fluid-flow tools of the unit, Bernoulli's equation and the continuity equation, in one tidy result. When you derive v = √(2gh), you're practicing exactly what the exam wants: identifying which terms in Bernoulli's equation cancel and justifying why (equal pressures, negligible surface velocity). It's also a favorite for the exam's reasoning style, since questions love asking how the exit speed changes if you double the depth, use a denser fluid, or widen the hole. If you understand the derivation, you can answer all of those without memorizing anything extra.

How Torricelli's Theorem connects across the course

Bernoulli's Equation (Unit 1)

Torricelli's Theorem is Bernoulli's equation with most of the terms crossed out. Set the pressure at the surface and at the hole both equal to atmospheric, treat the surface velocity as zero, and the leftover terms rearrange to v = √(2gh). If you can do that cancellation, you've basically derived the theorem from scratch.

Continuity Equation (Unit 1)

Continuity (A₁v₁ = A₂v₂) is what justifies ignoring the surface velocity. The tank's cross-sectional area is huge compared to the hole, so the surface creeps downward incredibly slowly. That 'wide tank' assumption is the hidden fine print in Torricelli's Theorem.

Incompressible Fluid (Unit 1)

The whole derivation assumes constant density, which is why ρ cancels out and the exit speed is the same for water, oil, or mercury. Density doesn't matter for exit speed, only the height of fluid above the hole does.

Velocity (Unit 1)

Once you have the exit velocity from Torricelli's Theorem, the escaping fluid becomes a projectile. A classic problem asks how far the stream lands from the tank, which means combining √(2gh) with horizontal projectile motion from your AP Physics 1 toolkit.

Is Torricelli's Theorem on the AP® Physics 2 exam?

No released FRQ uses the name 'Torricelli's Theorem' verbatim, but the setup shows up constantly: a tank with a hole, and you're asked for the exit speed, the landing distance of the stream, or how the speed changes when something about the setup changes. On multiple choice, expect ranking and comparison questions, like which hole produces the fastest stream or what happens to v when h is quadrupled (it doubles, since v scales with √h). On free response, the high-value skill is the derivation itself. Start from Bernoulli's equation, explicitly state that both points are at atmospheric pressure and that the surface velocity is negligible, then solve for v. Skipping those justifications is where points get lost. Also be ready for the trap answers, like thinking a denser fluid exits faster (it doesn't) or that a bigger hole means faster flow (bigger hole means more volume per second, not more speed).

Torricelli's Theorem vs Bernoulli's Equation

Bernoulli's equation is the general energy-conservation statement for flowing fluids, relating pressure, speed, and height between any two points. Torricelli's Theorem is one specific shortcut you get from it, valid only when both points are open to the atmosphere and the surface is nearly still. If a problem has a sealed pressurized tank or a fast-dropping surface, Torricelli's shortcut breaks and you have to go back to the full Bernoulli equation.

Key things to remember about Torricelli's Theorem

• Torricelli's Theorem says fluid exits a hole at v = √(2gh), where h is the height of the fluid surface above the opening.

• The exit speed equals the speed of an object free-falling from the fluid's surface to the hole, because the result is really just energy conservation.

• It's derived from Bernoulli's equation by assuming both the surface and the hole are at atmospheric pressure and the surface velocity is negligible (justified by the continuity equation in a wide tank).

• Fluid density and hole size do not affect the exit speed; only the depth h does, and speed scales with the square root of h.

• Since v depends on √h, doubling the depth multiplies the exit speed by √2, and quadrupling the depth doubles it.

• Exit-speed problems often chain into projectile motion, where you use √(2gh) as the horizontal launch speed of the escaping stream.

Frequently asked questions about Torricelli's Theorem

What is Torricelli's Theorem in AP Physics 2?

It's the result that fluid flows out of an opening in a container at speed v = √(2gh), where h is the height of the fluid surface above the opening. It appears in Topic 1.7 as a special case of Bernoulli's equation.

Does a denser fluid exit the hole faster according to Torricelli's Theorem?

No. Density cancels out of the derivation, so water, oil, and mercury all exit at the same speed for the same depth h. This is a common multiple-choice trap, since the exit speed depends only on g and h.

How is Torricelli's Theorem different from Bernoulli's equation?

Bernoulli's equation is the general relationship between pressure, speed, and height at any two points in a flowing fluid. Torricelli's Theorem is the special case where both points are at atmospheric pressure and the surface velocity is approximately zero, which collapses Bernoulli's equation down to v = √(2gh).

Why is the surface velocity assumed to be zero in Torricelli's Theorem?

Because of the continuity equation. The tank's cross-sectional area is much larger than the hole's, so A₁v₁ = A₂v₂ forces the surface to move extremely slowly compared to the exiting stream. That makes the surface velocity term in Bernoulli's equation negligible.

Does making the hole bigger increase the exit speed?

No. A bigger hole increases the volume flow rate (more fluid per second), but the exit speed still only depends on the depth h. Speed and flow rate are different quantities, and the exam likes testing whether you know the difference.