The Cartesian Diver: Teaching Buoyancy and Pressure the Way Archimedes Intended

Squeeze a plastic bottle and a small object inside it sinks. Release, and it floats back up. Your child just demonstrated three fundamental physical laws simultaneously — without realizing it. The Cartesian diver, named after philosopher René Descartes (though likely invented by Raffaello Magiotti in 1648), is one of the most physics-rich demonstrations you can make from a water bottle, a condiment packet, and sixty seconds of setup time. The reason it works involves Archimedes from 250 BC, Blaise Pascal from 1648, and Robert Boyle from 1662 — and the device in your hands also explains exactly how a submarine works.

Key Takeaways

• Archimedes’ principle: An object floats when its weight equals the weight of water it displaces. The diver floats when its average density is less than water’s (~1.0 g/cm³).
• Boyle’s Law: At constant temperature, pressure × volume = constant (PV = k). Squeezing the bottle increases pressure, which compresses the air pocket in the diver, reducing its volume.
• Pascal’s Law: Pressure applied to an enclosed fluid transmits equally in all directions — so squeezing the bottle transmits full pressure to the diver’s air bubble throughout the water.
• Net result: Compressed air = smaller volume = diver displaces less water = less buoyant force = diver sinks. Release pressure = air expands = diver rises.
• Real-world connection: Submarine ballast tanks use the same principle — flood tanks with water to sink, blow tanks with compressed air to surface.

Three Physics Laws in One Device

Archimedes’ Principle (250 BC)

Archimedes discovered that the buoyant force on an object equals the weight of the fluid it displaces:

F_buoyancy = ρ_fluid × V_displaced × g

Where:

• ρ_fluid = density of water = 1,000 kg/m³
• V_displaced = volume of water displaced by the object (m³)
• g = gravitational acceleration = 9.81 m/s²

An object floats when F_buoyancy ≥ weight of the object (m × g). This means it floats when its average density (total mass ÷ total volume, including any enclosed air) is less than or equal to the fluid’s density.

The Cartesian diver works because its average density is just barely less than 1.0 g/cm³ when correctly set up — the small enclosed air bubble is enough to tip the balance toward floating. A tiny change in that air volume tips it toward sinking.

Boyle’s Law (1662)

Robert Boyle established experimentally that for a fixed amount of gas at constant temperature:

P₁V₁ = P₂V₂

When you squeeze the bottle, the pressure inside increases (say, from 101 kPa atmospheric to 110 kPa). The air bubble inside the diver must contract proportionally:

V₂ = P₁V₁ / P₂ = (101 × V₁) / 110 ≈ 0.92 × V₁

The air pocket shrinks by roughly 8%. This seems small, but because the diver was calibrated to barely float, a small volume reduction causes a meaningful density change — enough to sink it.

Pascal’s Law (1648)

Blaise Pascal established that pressure in an enclosed fluid transmits equally throughout the fluid in all directions. This is why you don’t have to squeeze the bottle at exactly the same location as the diver — squeezing anywhere transmits the pressure increase to the diver immediately.

Pascal’s Law is also why hydraulic systems work: pressing a small piston creates pressure that, transmitted through fluid, can push a large piston with enormous force. Car brake systems, excavators, and aircraft control surfaces all run on this principle.

How the Diver Works Step by Step

1. Setup: A small object (ketchup packet, eyedropper, pen cap) with a small air bubble trapped inside floats near the surface of a full bottle of water.

2. Squeeze: Your hand compresses the bottle walls → Pascal’s Law transmits increased pressure throughout the water → The water pressure acts on the diver’s air pocket → Boyle’s Law compresses the air pocket → Volume decreases.

3. Sinking: The diver’s total volume decreases (less air, same water-filled mass) → Average density increases → Archimedes’ principle: buoyant force now less than weight → Diver sinks.

4. Release: Pressure drops → Air pocket expands → Volume increases → Average density decreases back below 1.0 g/cm³ → Buoyant force exceeds weight → Diver rises.

The elegance of this device is that you’re directly controlling density through pressure — no changes in mass, just a reversible volume change.

Diver Designs Compared

Diver Type	How Air Is Trapped	Sensitivity	Setup Difficulty	Best For
Ketchup packet	Pre-existing air bubble in packet	Easy to calibrate	Very easy — just test multiple packets	Quickest demo; consistent results
Eyedropper (medicine dropper)	Air trapped in rubber bulb + glass tube	Highly adjustable	Moderate — squeeze to set water level	Best for precise hovering demonstrations
Pen cap	Air trapped at closed end of cap	Moderate	Easy — drop in with small amount of clay	Good for testing shape effects
Test tube + clay	Air trapped in tube, clay as ballast	Adjustable	Moderate	Good for quantitative pressure experiments
DIY (folded aluminum foil boat)	Air trapped in folded enclosure	Difficult to calibrate	Harder	Good for design challenge extensions

The ketchup packet trick: Not all ketchup packets float — you need one that barely floats (the air bubble is just enough to displace water equal to the packet’s weight). Fill a tall glass with water and test several packets. Packets that sink fast have too little air; packets that float high have too much. Find one that barely floats with slight encouragement. That’s your diver.

Submarine Ballast Tanks: The Real-World Version

A submarine does exactly what your Cartesian diver does, just with controlled flooding of large ballast tanks instead of air pocket compression:

To dive: Open ballast tank vents → seawater floods in → submarine’s average density exceeds 1.025 g/cm³ (seawater) → submarine sinks. Control planes (small rudder-like fins) then control depth and angle.

To surface: High-pressure compressed air (stored in tanks at ~200+ atmospheres) blows water out of ballast tanks → average density decreases below seawater → submarine rises.

Emergency surfacing: Even if all systems fail, submarines have emergency air reserves that can blow tanks in seconds, bringing the sub to the surface automatically — the same physics, just faster.

The U.S. Navy’s Virginia-class submarines displace about 7,900 metric tons when surfaced. The ballast tanks can hold enough seawater to change this displacement by thousands of tons — a Cartesian diver that’s 115 meters long.

Other applications of the same physics:

• Fish swim bladders: Fish inflate/deflate a gas-filled bladder using a rete mirabile (counter-current gas exchange system) to adjust buoyancy — biological Cartesian divers.
• Scuba BCD (buoyancy compensator device): Divers add or release air from an inflatable jacket to control depth — identical mechanism.
• Weather balloons: As altitude increases, atmospheric pressure decreases, and the helium in the balloon expands (Boyle’s Law) until the balloon reaches equilibrium buoyancy.

How to Teach Your Kid About the Cartesian Diver

Ages 5–8: Ketchup Packet Diver

Materials: A 1.5 L or 2 L plastic bottle with cap, water, multiple Heinz or McDonald’s ketchup packets (test several to find one that barely floats).

Fill the bottle completely with water — completely, no air space inside. Add the floating ketchup packet. Screw the cap on tightly. Now squeeze the bottle. The packet sinks. Release — it floats. Let your child do this 20 times. It never gets old.

Discussion questions: “What’s inside the packet? What’s in the packet that’s different from the water around it?” (Air.) “What do you think happens to the air when you squeeze?” (It gets squished.) “Why does squishing the air make the packet sink?” Work toward: smaller air pocket = less floating power.

The question to ask: “If we made a tiny hole in the packet so all the air escaped, what would happen when you squeezed?”

Ages 9–12: Design a Hovering Diver

Materials: Plastic bottle (2L), water, eyedropper (medicine dropper), modeling clay, ruler.

Goal: Make the diver hover at exactly the middle of the bottle — not touching the top, not touching the bottom — with moderate squeezing.

Fill the bottle. Squeeze some water into the eyedropper so it just barely sinks when dropped in. Add a tiny bit of clay to the nozzle for weight adjustment. Experiment until you find the balance where light squeezing sends it down, releasing brings it back up, and medium squeezing holds it stationary in the middle.

Science journal: Record: how much water is in the dropper when it barely floats vs. sinks. Try to calculate the volume of the trapped air at different squeeze pressures. If you know the depth of the diver and the density of water, can you estimate the pressure increase at the bottom vs. top of the bottle (hint: every 10 cm of water = 0.01 atm additional pressure)?

The question to ask: “What would happen to your diver if you used saltwater instead of freshwater — would it need more or less air to barely float? Why?”

Ages 13+: Quantitative Boyle’s Law Verification

Materials: 2L bottle, pressure gauge (available as bicycle tire gauges or smartphone apps using the barometric sensor), ruler, eyedropper diver, calculator.

Experimental design:

1. Measure the initial volume of air in the eyedropper (by filling with water and measuring water displaced, then subtracting from total dropper volume).
2. Mark water levels on the dropper at measured volumes.
3. Apply known pressures by squeezing (this is the hard part — estimate pressure by the deflection of the bottle wall, or use a pressure gauge on the cap if you can drill a port).
4. For each pressure level, measure the compressed volume of air in the dropper.
5. Plot P vs. 1/V (should be linear if Boyle’s Law holds) or PV vs. pressure (should be constant).

Alternative quantitative approach: At what depth would the diver need to be submerged in the open ocean (no bottle) to compress its air bubble by the amount you observe? Use the hydrostatic pressure formula: P = ρgh, where ρ = 1000 kg/m³, g = 9.81 m/s², h = depth in meters. Calculate h.

Connect to submarines: Research the crush depth of different submarine classes (publicly available). Calculate the ratio of surface pressure to crush-depth pressure. What does Boyle’s Law predict happens to any gas pockets at crush depth?

The question to ask: “At what ocean depth would a hollow steel sphere (with a fixed air pocket inside) have its buoyancy exactly neutralized by the compressed air? What does this tell you about the design challenges for deep-sea submersibles?”

What to Watch For Over 3 Months

• Week 1: Does your child explain the diver using the word “squished” or “compressed”? Both are right — compressed is the scientific term, but the concept matters more than vocabulary at this stage.
• Month 1: Do they apply the concept elsewhere? “Is that why the basketball gets harder to squeeze when it’s fully inflated?” (Yes — Boyle’s Law.) “Is that why ears pop on airplanes?” (Yes — pressure change.)
• Month 2: Can they predict what would happen if you used a thicker-walled bottle (requires more force, same physics), or if the water were warmer (affects gas behavior slightly)?
• Month 3: The ultimate indicator — they can explain submarines using Cartesian diver physics without being prompted, and they ask about the bends (nitrogen narcosis) in scuba diving, which is the next chapter of gas-under-pressure biology.

Frequently Asked Questions

Why does the bottle need to be completely full with no air space? If there’s an air space in the bottle, squeezing compresses that air bubble instead of transmitting pressure through the water to the diver. Water is nearly incompressible, so a full bottle transmits pressure directly; a bottle with an air gap is “squishy” in the wrong way.

What if the diver just sinks to the bottom and never comes back up? Your diver is too dense — it doesn’t have enough air trapped to float at all when pressure is released. Try a different ketchup packet, or if using an eyedropper, squeeze some water out so more air remains inside.

Why does it work better with a tighter cap? A loose cap lets water escape when you squeeze, reducing pressure transmission. A tight seal ensures all your squeezing force goes into pressurizing the water, which then compresses the diver’s air bubble.

Could you make a Cartesian diver that works with air pressure instead of water pressure? In principle, yes — if you suspended a balloon inside a larger inflatable container and changed the pressure in the outer container. However, air is already compressible (unlike water), which makes calibration much harder. Water’s near-incompressibility is what makes the Cartesian diver work so elegantly.

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About the author

Ricky Flores is the founder of HiWave Makers and an electrical engineer with 15+ years of experience building consumer technology at Apple, Samsung, and Texas Instruments. He writes about how kids learn to build, think, and create in a tech-saturated world. Read more at hiwavemakers.com.

Sources

1. American Association of Physics Teachers (AAPT). (2023). “Buoyancy and fluid pressure demonstrations for K-12.” The Physics Teacher, 61(4), 256–261.
2. Boyle, R. (1662). New Experiments Physico-Mechanicall, Touching the Spring of the Air and Its Effects. Oxford: H. Hall for T. Robinson.
3. Hewitt, P. G. (2021). Conceptual Physics (13th ed.). Pearson. Chapter 14: Liquids, pp. 261–295.
4. U.S. Navy Submarine Medical Research Laboratory. (2021). “Submarine ballast systems and hydrodynamics.” NavSea Technical Publication.
5. National Science Teaching Association (NSTA). (2022). “Cartesian diver: A classic demonstration with modern physics analysis.” The Science Teacher, 89(3), 44–49.
6. Pascal, B. (1663). Traité de l’équilibre des liqueurs. Paris. (Foundational text on hydraulic pressure.)