In the original Pirates of the Caribbean, Captain Jack Sparrow and Will Turner escape Port Royal by strolling out to an anchored ship. They do this by walking on the seafloor, using an upside-down rowboat to hold air so they can breathe. It’s an awesome scene, but is it, er, possible? Could you do this yourself?
Of course, the great thing about movies is that they don’t need to be realistic (and the Pirates franchise makes the most of that exemption). I’m only asking about this because it’s a fun way to do some physics and understand how the world works. So grab your goggles and let’s plunge in!
What Floats Your Boat?
It may not seem like it, but objects in the water still have gravity acting on them, same as on land. Then why do you feel weightless underwater? Well, let’s back up for a moment. Imagine you have a block of steel and a block of styrofoam, both measuring 1 cubic foot in volume. Each one has a mass, which equals its volume (V) times its density (ρ). Then the gravitational force (Fg) on each is simply mass times the gravitational field (g):
But you already know about this, because Fg is what normies call an object’s “weight,” and for a given volume, weight depends only on the density. Now, if you dropped these blocks in a lake, obviously the styrofoam would float and the steel would sink. So clearly it has something to do with density.
What if you had a block of water with the same volume? If you could somehow hold this cube of water, it would feel pretty heavy, about 62.4 pounds. Now, if you place it carefully in a lake, will it sink or bob on the surface like styrofoam? Neither, right? It’s just going to sit there.
Since it doesn’t move up or down, the total force on the block of water must be zero. That means there has to be a force counteracting gravity by pushing up with equal strength. We call this buoyancy, and for any object, the buoyancy force is equal to the weight of the water it displaces.
So let’s think about this. The steel block displaces the same amount of water, so it has the same upward-pushing buoyancy force as the block of water. But because it’s denser and has more mass, down it goes.
In general, an object will sink if the gravitational force exceeds the buoyancy force, and it will float if the buoyancy force exceeds the gravitational force. Another way of saying that is, an object will sink if it’s denser than water and it will float if it’s less dense.
And right in the middle an object will neither sink nor rise to the surface—we call that neutral buoyancy. Humans are pretty close to neutral because our bodies are 60 percent water. That’s why you feel weightless underwater—the buoyancy force pretty much offsets the gravitational force.
Avast! Hold on there, matey. Aircraft carriers are made of steel and weigh 100,000 tons, so why do they float? Can you guess? It’s because of their shape. Unlike a block of steel, a ship’s hull is hollow and filled with air, so it has a large volume relative to its weight.
But what if you start filling it with cargo? The ship gets heavier, which means it must displace more water to reach that equilibrium point. In general, when you launch a boat or ship into the water, it’ll sink down until the weight of the water it pushes aside equals the boat’s total weight.
Cool, right? This was all figured out by Archimedes more than 2,000 years ago, as legend has it, when he stepped into a bathtub and saw the water level rise.
Forces on an Underwater Boat
So, what would you need to walk an upside-down boat on the bottom of the ocean? Let’s start off with a force diagram:
There are four forces on this boat: There’s the upward-pushing buoyancy force (FB) and the downward-pushing gravitational force (mass times the gravitational field, mg). Finally, Will and Jack have to push up on the boat so the boat pushes down on them. I mean, it’s pretty hard to walk on the bottom when you’re effectively weightless. (Trust me, I tried.)
So let’s do the math. The forces from the humans are puny; we can ignore those. For buoyancy, we find the weight of the water displaced, using the same formula we used for gravity above, F = ρVg. The density (ρ) of water is 1,000 kg/m3, and let’s say the air-filled boat has a volume of 3 m3. Multiplying this by the gravitational field g, which is 9.8 newtons per kilogram, we get a buoyancy force of 29,400 newtons. In imperial units, which is probably what Jack and Will use, that’s 6,600 pounds.
What does this mean? It means the boat must weigh at least 6,600 pounds—which it clearly doesn’t—or the buoyancy will rocket them right up to the surface. I see only two solutions: They could let the air out to reduce the volume—obviously not a first choice—or they could add more than 3 tons of ballast. Maybe gold doubloons? Pirates like gold, and it’s extremely dense. But it’s hard to be stealthy when you’re running wheelbarrows of gold down to the beach.
What About Air Compression?
I see a hand in back. Isn’t it true that the air inside would get compressed the deeper you go, which would reduce the volume and thus reduce the buoyancy? Smart question. Yes, that would indeed happen since the water pressure increases with depth. But would it be enough?
To figure this out, we can use the ideal gas law. This gives us a relationship between the pressure (P) of a gas and its volume (V):
In this model, n is the amount of gas (in moles), and T is the temperature. (R is just a constant.) As the boat moves down into deeper water, the amount of air trapped inside doesn’t change, and for simplicity let’s assume the temperature is constant too.
That means the whole right side of the equation is constant. Which means, in turn, that pressure and volume are inversely proportional: If P rises, V shrinks—and vice versa. That’s Boyle’s law.
Now, if you’re submerged, the pressure is the sum of the atmospheric pressure and the water pressure. That might seem weird, but it’s because you have the weight of the atmosphere above the surface (1 atm) weighing down on you as well as the weight of the water column above you. As a rule of thumb, this total pressure increases by about 1 atm for every 10 meters you descend.
Call the pressure and volume at the surface P1 and V1, and for simplicity let’s say the starting volume of air in the boat is 1 m3. We’ll call underwater pressure and volume P2 and V2. Now, with our assumptions, Boyle’s law says P1V1 = P2V2. So we can easily solve for the volume of air at a depth of, say, 5 meters: P1V1/P2 = 1/1.5, so V2 is 0.67 m3.
In other words, the volume declines by one-third. But that still gives us a buoyancy force of 4,400 pounds. If Jack and will pulled the boat down to a depth of 30 meters (about 100 feet), the volume and buoyancy would decrease by 75 percent. But that’s still a buoyancy force of 1,650 pounds, and it’s way too deep. Even if you could get down there, breathing compressed air at 4 atm’s would require a very slow, controlled ascent to avoid the bends.
Bottom line: This trick for walking underwater is pure fantasy—as phony as fool’s gold—but, hey, it’s 24-karat entertainment.
The post Could You Use a Rowboat to Walk on the Seafloor Like Jack Sparrow? appeared first on Wired.
