What Is Torricelli’s Law? | The Tank-Drain Rule That Works

It states that a liquid’s exit speed from a small hole matches the speed an object gains falling through the same vertical drop.

Torricelli’s law is one of those physics ideas that feels like a magic trick the first time you use it. You look at a tank of water, punch a tiny hole in the side, and suddenly you can estimate how fast the jet shoots out using only the height of water above the hole. No fancy pump curves. No long equations. Just gravity doing what gravity does.

This matters in real work: lab setups, rain barrels, aquarium plumbing, water towers, DIY fountains, irrigation buckets, even quick sanity checks in fluid mechanics homework. If you can estimate the exit speed, you can estimate how far a stream will travel, how long a container will take to drain, or how much flow you can expect through an opening.

What Torricelli’s Law Says In Plain Words

Take a tank with a small opening on its side. Let the water surface sit a vertical distance h above the center of that opening. Torricelli’s law says the water leaving the hole has a speed:

v = √(2 g h)

Here’s the simple meaning:

• v is the exit speed of the fluid (meters per second).
• g is gravitational acceleration (about 9.81 m/s² near Earth’s surface).
• h is the vertical head: the height of liquid above the hole’s center (meters).

So if the water level drops, h drops, and the jet slows down. That’s why a container drains fast at first and then crawls near the end.

Why The “Falling Drop” Idea Fits So Well

Torricelli’s law connects fluid flow to a familiar motion: a falling object. If you drop a pebble from height h, it picks up speed as it falls. Neglect air drag and you get the same speed relationship, v = √(2 g h).

In the tank, a tiny packet of water near the surface can’t literally fall straight down through the tank wall. Still, energy changes in a similar way. As water moves from the surface region down to the hole region, it loses gravitational potential energy. That energy shows up as kinetic energy in the jet.

That mental picture gives you a fast gut-check: doubling the height doesn’t double the exit speed. It raises speed by the square root factor. So a tank with four times the head gives about twice the speed.

Torricelli’s Law In Fluid Flow Calculations

Most textbooks connect Torricelli’s law to Bernoulli’s equation. The bridge is clean: along a streamline, energy per unit volume stays consistent when viscosity effects are small and flow is steady. If you take one point at the free surface and one point just outside the orifice, the pressure terms cancel (both are at atmospheric pressure), the surface speed is often treated as near-zero in a big tank, and the height difference becomes the driver.

If you want to read the formal statement and the classic form of the equation, Encyclopaedia Britannica’s entry on Torricelli’s law lays it out in a compact way. If you want the broader energy relationship behind it, NASA Glenn’s explanation of Bernoulli’s equation is a clear reference that matches what students see in intro fluids.

In practice, people use Torricelli’s law in two modes:

• Back-of-the-envelope estimates: “Is this jet going to shoot across the sink or just dribble?”
• First-pass design: “What orifice size and head give me the flow range I need?”

Once you have the speed, you can turn it into flow rate with the opening area. For a circular hole, area is A = π(d/2)². A basic flow estimate is:

Q ≈ A v

Real liquids also bring losses at the opening, so engineers often multiply by a discharge coefficient Cd:

Q ≈ Cd A √(2 g h)

That coefficient bundles effects like jet contraction and viscous loss near the edge. You’ll see it used in lab work and hydraulic design.

What Each Symbol Means And How To Measure It

Most mistakes come from measuring h in a sloppy way. The height h is the vertical distance from the water surface to the centerline of the hole (not to the bottom of the tank, not to the ground, not to the top rim).

Also, keep units clean. If h is in meters and g is in m/s², then v comes out in m/s. If you measure h in centimeters, convert it first.

If the opening is a short tube or nozzle, the “effective” spot where speed forms can shift, and losses rise. In that case, the coefficient matters more than the pure Torricelli speed.

Quantity You Use	What It Represents	Practical Note
h (head)	Surface height above the orifice center	Measure vertically; use the center of the hole as the reference point
g	Gravitational acceleration	Use 9.81 m/s² for many Earth-based problems
v	Jet exit speed	Applies right at the exit, before breakup into droplets
A	Area of the opening	For a round hole: A = π(d/2)²
Q	Volumetric flow rate	Often estimated as Q ≈ A v for a first pass
Cd	Discharge coefficient	Typical lab values for sharp-edged orifices often fall near 0.6–0.65
Sharp-edged orifice	A thin wall opening with a crisp edge	Tends to create a contracted jet, so Cd matters
Large tank assumption	Surface speed treated as near-zero	Works best when tank cross-section dwarfs the hole area

Where The Law Works Well And Where It Trips People Up

Torricelli’s law shines when the setup matches its built-in assumptions. You don’t need perfection, but you do need the setup to be “close enough” that gravity-driven energy change is the main story.

Setups That Usually Match Well

• A big container with a small, sharp-edged hole
• Water or another low-viscosity liquid
• Short discharge into open air
• Steady draining where the water level drops slowly compared with jet speed

Setups That Need Extra Care

• Thick liquids: syrup, oil, paint. Viscous loss rises fast.
• Long tubes: friction along the tube can dominate the flow.
• Small holes: surface tension and viscosity start to bully the ideal model.
• Pressurized tanks: added pressure above the surface adds extra driving head.

If the tank is sealed and pressurized, the exit speed can exceed √(2 g h) since pressure energy joins the party. If the tank is under partial vacuum, the opposite happens.

How To Use It For Real Estimates

Here’s a reliable workflow that stays simple and still gets you usable numbers.

Step 1: Measure The Head

Measure the vertical distance from the liquid surface to the center of the opening. Use a ruler or a marked stick. If the surface is sloshing, wait for it to settle.

Step 2: Compute Exit Speed

Use v = √(2 g h). Keep units consistent.

Step 3: Turn Speed Into Flow Rate

Compute the opening area A. Multiply: Q ≈ A v. If you’re working with a sharp-edged orifice and you want a closer match, multiply by Cd too.

Step 4: Sanity-Check The Result

If your flow rate feels off, check the usual culprits: wrong h reference, wrong diameter, unit slip, or a setup that adds friction (like a tube) near the hole.

Drain Time: Why It’s Not Just “Volume Divided By Flow”

People often try: “My tank is 20 liters. My flow is 1 liter per second. So it drains in 20 seconds.” That only works if flow stays constant. A draining tank doesn’t do that. As the surface drops, h drops. Then v drops. Then Q drops.

Still, you can get a strong estimate without heavy calculus by splitting the drain into slices. Pick several water levels, compute the flow at each level, then average across short intervals. It’s a spreadsheet-friendly method and it mirrors what many lab manuals do.

Here’s a practical way to do it:

1. Mark 5–10 height levels on the tank wall.
2. For each level, compute v and then Q.
3. Estimate how much volume sits between each pair of marks.
4. Time for each slice is roughly (slice volume) ÷ (average flow for that slice).
5. Add slice times for total drain time.

This keeps the physics honest while staying friendly to real measurements.

Task You Want	What To Calculate	Handy Tip
Exit speed from a hole	v = √(2 g h)	Use the hole’s centerline for h
Flow rate through a round hole	Q ≈ Cd A √(2 g h)	If you lack Cd, start with 0.6 for sharp edges
Jet range to a lower level	Combine exit speed with free-fall time	Measure vertical drop to the landing level to get fall time
Compare two tank heights	v scales with √h	Four times the head gives about twice the speed
Drain-time estimate	Slice-by-slice flow average	More slices give a tighter estimate with the same formula set
Spot a mismatch in a lab	Compare measured Q to Torricelli-based Q	Large gaps point to friction, viscosity, or a non-sharp outlet

Common Misreads That Lead To Wrong Answers

Torricelli problems look easy, so small slips can sneak in. Here are the big ones that show up again and again.

Using The Wrong Height

The height is not “water depth in the tank.” It’s the vertical distance from the free surface to the hole center. If the hole sits halfway up a tall tank, h can be small even when the tank still holds a lot of water.

Forgetting Jet Contraction

With a sharp-edged hole, the jet narrows just after exiting. That shrink reduces effective area, which reduces flow rate compared with Q ≈ A v. That’s one reason Cd exists.

Mixing Up Speed And Flow

A bigger hole does not change the ideal exit speed for a given head. It changes the flow rate because area changes. Speed is set by head; volume per time is set by head and opening size.

Applying It To Long Pipes Without Adjustments

Once the water must travel through a long tube, friction can dominate. You can still start with Torricelli as a first pass, then bring in head-loss models if you need a closer match.

How Students Can Explain It Clearly In Assignments

If you need to write Torricelli’s law in a report or exam, keep it crisp:

• State the law: exit speed depends on head, v = √(2 g h).
• Define h as the vertical distance from the surface to the orifice center.
• State the main assumptions in plain language: gravity-driven flow, small outlet, losses small or handled by a coefficient.
• If you compute flow, show the step from speed to Q using area.

That structure shows you know what the equation means, not just how to punch buttons on a calculator.

Mini Checklist For Using Torricelli’s Law Without Regret

• Measure head from surface to hole center, straight up and down.
• Keep units consistent from start to finish.
• Use v = √(2 g h) for exit speed.
• Use Q ≈ Cd A √(2 g h) when you need flow rate.
• If the outlet is a tube, expect friction to cut flow below the ideal value.
• If the tank is pressurized, expect higher speeds than gravity-only predicts.