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So we have the tragedy of the submersible being lost near the Titanic site.

I am trying to understand the pressure the vessel was under. I have heard the "$1$ atmosphere for every $10$ meters" approximation. So if the implosion happened $1$ $km$ down, $100$ atmospheres of pressure were acting on the sub.

Is it also valid to think of the column of water on top of the sub? If the sub is $20'\times7'$, and $0.6$ miles down, this comes out to $28.3$ $million$ pounds ($12.9$ $million$ $kg$) of water bearing down on the hull.

My numbers may not be right, but they aren't crazy - there's a lot of water on that sub. How would a physicist explain the force of the implosion to a non-physicist in a way that conveys the effect?

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TLDR: To understand it, don't think about pressure. Think about energy.

The best approach here is to think in terms of energy/work, instead of pressure. There's lots of situations that become more intuitive if you try tothink in terms of energy, instead of other physical measures.

For example, people have notoriously bad intuition about the dangers of driving above the speed limit. If driving 70 km/h is OK, then surely driving a bit faster at 100 km/h can't be that bad, they think. But if you reframe the situation in terms of energy, it becomes clear why speeding is a bad idea. Kinetic energy is calculated as $\frac{mv^2}{2}$, so it scales with the square of velocity. So, in going from 70 km/h to 100 km/h, the kinetic energy increases by $\frac{100^2}{70^2} \approx 2.04$. So a increase of less than 50% in velocity more than doubles the collision energy. The degree you end smashed in an accident is proportional to collision energy, not just to velocity, and a small difference in terms of km/h in speed may be the difference between dying on the spot or walking out of the wreck in your own feet.

The same way, thinking in terms of pressure alone is meaningless. The pressure at those dephts is very high, but there are living creatures that live on even greater depths with no ill effects. To explain this difference, we must calculate the energies involved in each situation.

Energy can be estimated from pressure and volume, as $E = P * \Delta V$, as pressure has units of $\frac {Force}{Area}$, and volume has units of $Area*Distance$, so $\frac {Force}{Area}*Area*Distance = Force*Distance$, that is the definition of work, with the same units as energy. Water has very low compressibility, so despite the huge pressures abyssal fish thrive in, the volume differences they are likely to face are negligible, so energies calculated from the resulting product are small.

On the other hand, the submersible crew were not inmersed in water, like the fish. They inhabited a very compressible gas bubble, shielded against the surrounding ocean pressure by the submersible hull. Wikipedia gives the inner dimensions of the cylindrical hull as 2.4m lenght and 1.42m diameter (0.71m radius), so the volume of the gas bubble was approximately $\Pi*0.71m^2*1.42m = 3.8m³$.

The Titanic wreck sits in the ocean floor about 3800m deep. The submersible descent was expected to last 2h. They lost contact after 1h45min. So we can estimate the implosion happened about $\frac{105min}{120min}*3800m = 3325m$ deep.

There is a rule of thumb every 10m water depth adds the equivalent about one atmospheric pressure. One atmospheric pressure is about $10^5$ Pascals. So we can estimate the pressure differential at implosion moment as $332.5 * 10^5 = 3.325 * 10^7 Pa$.

Now with estimates for the water pressure and gas volume bubble, we can calculate a rough estimate for implosion energy as follows:

$$E = P * \Delta V = 3.325 * 10^7 Pa * 3.8m^3$$ $$E = 126.4 megajoules$$

This is a large amount of energy. For comparison, one conventional Kg TNT equivalent is 4.184 megajoules. So this implosion had the energy equivalent to a charge of $\frac{126.4}{4.184} \approx 30 kg$ of high explosives.

With this information we have a scenario that is more intuitive to grasp by a layperson. We can imagine an equivalent situation in what each metal cap at the ends of the composite tube that comprised the submersible are projectiles, each connected to a cannon loaded with an explosive charge equivalent to about 15kg of TNT. Then the charges went off simultaneously, as the shoddily made central tube collapsed:

enter image description here

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Yes, you are right that we can calculate the force on the sub by adding up the total weight of everything above the submarine (the water), this force is

"Force" = "Area of Sub" $\times$ "height of water column" $\times$ "density of water" $\times$ "little g"

The pressure is the force per unit area (the total force divided by the area it is spread over). So for pressure we divide the force by "Area of the Sub", so that leaves

"Pressure" = "height of water column" $\times$ "density of water" $\times$ "little g"

The density of water is ~1,000 kg/m$^3$. Little g is ~10 m/s$^2$. 1 Atmosphere (the pressure in Earth's air at sea level) is about 10,000 kg/ms$^2$. This gives us back that 10 meters = 1 atmosphere rule of thumb.

The depth of the Titanic is apparently about 3,800m. That gives a pressure of about 381 atmospheres. (380 atmospheres from the weight of the water in a column above the sub, and another 1 atmosphere from the weight of the air in that same column going up to the edge of space.)

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  • $\begingroup$ There's also a (fairly minimal) contribution from the atmosphere too. Your expression is for the gauge pressure, not the absolute pressure. $\endgroup$
    – hft
    Jun 23, 2023 at 18:35
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    $\begingroup$ @hft : You missed the last paragraph where the atmosphere is added on. But yes, that is very easily forgotten. $\endgroup$
    – Dast
    Jun 28, 2023 at 9:57

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