TL;DR Cliffs:
1) Potential energy doesn't equate to transfer of that energy as trauma to a target.
2) What helps transfer energy that would be over penetration is spreading the energy over a large an area as possible while imparting as much inertia through spinning.
3) 9mm hollow points spread on average just about as wide as .45 acp
4) 9mm rounds spin twice as fast as .45 acp rounds.
5) .45 acp rounds have more potential energy, not a large amount more for what is needed to cause damage, but are inefficient at imparting that energy as trauma to a target with ballistic gel density.
6) Because .45 acp is inefficient as imparting it's greater potential energy as trauma to a target compared to 9mm which is more efficient, the wound channels and total damage caused tends to be the same.
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IrishScott,
What your math doesnt' tell you is the transference of kinetic energy over what area. It takes exactly 9 pounds of pressure to usually puncture human skin. If I apply that pressure through a needle point I'm going to poke skin If I apply that same pressure using a coke can I'm going to just get an indention in the skin. It pressure required for a coke can to puncture skin like a needle is significantly more since the can spreads the pressure out. So say it takes 500 pounds of pressure for a coke can to puncture the skin. You can compare that to 500 pounds of pressure to the needle. The can is going to do significant more damage than the needle at the same amount of force being applied.
Meaning it's not a complete static scenario. In math a .45 acp bullet moving a 800 fps is going to have more kinetic force than a 9mm bullet moving at 1059 fps. In ball ammo with the spread of force from the bullets being similar in cross section, the damage is going to be greater with the .45 acp. The difference is changing HOW that force is applied and how to harness more of that kinetic energy into the surrounding tissue to cause more trauma.
The problem becomes one of transference. Not potential energy. Both a 9mm and .45 acp round in ball ammo has enough force and energy to completely pass through and over penetrate the human body. Both will penetrate over 14" in human organ thickness ballistic gelatin. This is far enough to enter the gut of the average human and out the back and still keep going. Which means effectively that the potential kinetic energy the bullet has is not being fully imparted upon the target.
The goal of a defensive round is to cause as much trauma as possible with a single round. Ideally, you'll want all the potential energy to be imparted into the target. Which means have the bullet enter the target and stop just short of exiting. That is the most efficient usage of the round. With ball ammo, the only way to do that is to use less powder to impart less energy. But doing so doesn't make the wound any better just because the bull stops just shy of exiting the target because it lacks the power to push through. It's better to have the bullet expand to create larger surface area to put that pressure it has over a greater surface and cause more trauma. Just like the coke can example I talked about earlier.
The other key factor in imparting more energy into a target is spinning the tissue as it is being caught by the bullet. This imparts more inertia into the tissue causing it to pull tissue with it when being pulled on by the hollow point bullet.
And you have to consider the amount of force needed to cause trauma to a target in the first place. One can point to math and say, see .45 acp has X amount more potential energy than 9mm, but that doesn't mean squat when looked at in comparison to the energy needed to cause certain levels of trauma based on the material resistance of the target in question to energy being directed at it.
For math with an analogy to this would be the following. 1 grain is equal to 64.8 milligrams. So a 147 grain 9mm bullet weights in at 9.525 grams. A 230 grain .45 acp weighs in at 14.904 grams. For the standard F=MA equation that uses kg for mass, that is .009525 kg and .014904 kg. Using average velocities of non +p ammo from the previous testing site I link, the average 147 gr had an average muzzle velocity of 1000 fps or 304.8 m/s as recorded. It's a little faster than SAAMI specification, but within tolerances.
The average 230gr .45 acp was about 850 fps. Again a little fast, but in tolerances. So 266.7 m/s for the .45 acp.
Using handy dandy physics formula we can see that the 9mm is...
(.009525 kg)(304.8 m/s) = 2.90322 N
and the .45 acp is...
(.014904 kg)(266.7 m/s) = 3.9748968 N
But we already know that the .45 acp rounds have more energy. The little example math both irishScott and I did show that. The problem is how to impart all that energy into trauma on a target. The most efficient way is to impart as much force over an area as large as possible while spinning as fast as possible to provide inertia into the target. Both are key.
A 9mm by design is a bit more elongated. But lets look at the average spread of the hollow points that were recovered from the previous site throwing out the highest and lowest value as outliers. Which would be 0.356 inches for smallest and 0.821 inches for the biggest. We have an average spread of 0.640 inches.
Same thing with the .45 acp. Throw out the highest of 0.987 and the and the lowest outlier with a spread of 0.451 inches. Then the average was about 0.687ish inches for spread. Bigger spread but not as big as one might think considering the initial diameters of the bullet.
With the twist rate for a 230gr .45 acp at an average of 1 in 16" we get this for revolutions per minute using our current stats.
850 ft./sec X 1 rev./16 inches X 60 sec./min X 12 in./ft. = 38250 rev./min
The spin of the average 147gr 9mm with a 1 in 10" twist rate is.
1000 ft./sec X 1 rev./10 inches X 60 sec./min X 12 in./ft. = 72000 rev./min
So the 9mm is spinning twice as fast basically than the .45 acp. That spinning helps impart more kinetic energy to the target. I really don't know any formula to take the twist rate, ballistic gel density, potential energy and petal spread into account to determine potential trauma over a given length. I will state while the .45 acp has more potential energy, the fact that it barely spreads bigger than the 9mm as a hollow point and has a much lower spin rate means that less of it's energy is imparted as trauma onto surrounding tissue in a target than a 9mm with current design setups of modern hollow point bullets.
Remember, it's not how much potential energy that a caliber has, but how much of that energy it can impart on to the largest area possible of a target without wasting the energy. This is why real world testing of modern 9mm hollow points are very close in trauma caused to .45 acp hollow points. The difference as shown by real world testing proves that. The diameters of the wound channels created is about the same on average. Some 9mm shots have bigger wound channels and some have smaller.