I'm not really equipped to break down all the physics involved.
But I can assure you that a 45 auto is not comparable in damage to the 7.62, even from the short barrel.
The 45 is not that slow a round either, except as compared to rifle rounds.
It is a more than adequate round for a handgun though.
If it fails to penetrate a target, it is due to it's much larger frontal area than a 9mm, and lack of power.
Transfer of force must be ascribed to that surface area.
When we tested a self defense 45 round, the increase in power was amazing over a standard round.
Very impressive.
It also expanded fully in a few inches of water.
But it did not compare to a run of the mill 7.62 x 39 fired from a 16" barrel.
The wider base has a better burn characteristic that lends itself to longer barrels too.
The diameter of the cartridge is it's benefit and disadvantage.
It slows down very rapidly in the air, losing power and accuracy.
More importantly, it takes up space, in the weapon, the mechanism and so on.
Because the weapons tend to be heavier, and the firing characteristics are different, I find the 45 very easy to fire.
I think it's the most effective round among common handgun sizes.
There are some unusual rounds with more power than a 12 gauge shotgun shell, like the 454 Casull, but the usual questions come up of gun handling, rate of fire, physical size and so on.
I don't know if something different is happening with these different rounds, or if it all balances out as physical force.
I suspect that it may be like the difference in being hit with a hammer or run through with a spear.
I don't have any numbers, but the percentage of force transferred was dramatically higher with the rifle.
Since it has about 300% of the power of the 45, it isn't too unexpected.
We wanted to know how it worked, so we tried it.
You should too.
Both rounds went straight through, transferring power to the water in the process.
It is notable how effectively the 45 expanded in only water and thin plastic.
I don't think this was due to a particular round.
I think the wider round slows more rapidly and this ensures flattening.
There are DPX 7.62 rounds and fragmenting Glaser rounds too.
So spreading of force can be enhanced this way too.
The Glaser should help in penetration issues, or at least any ricochets.
As for the shotgun, I'm guessing you're referring to the 12 gauge?
These use as much powder as 1.5 308 rounds.
They are very inefficient, but produce a lot of force for a short range.
I think total power transferred to a target is about the same as the short barreled 7.62.
More than adequate usually.
There are some disadvantages in spreading force out, and some advantages.
A shotgun is what I recommended to my friend with the SKS rifle.
Plenty of different ammo choices and fairly idiot proof.
The best choice for most people going beyond a handgun.
I do think people are too confident about not needing to be accurate.
You still have to hit what you aim at.
I do think a rifle is more likely to drop someone, but up close it really doesn't matter.
I saw a documentary on war or something and they were discussing how through and through wounds have changed.
The same apparent wound becomes much more deadly with increases in power.
And difficult to treat.
The high force transferred in a seemingly clean wound path does a lot of damage.
Perhaps it is comparable to the deer you mentioned hitting and found the meat to have been turned to mush?
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Over Penetration / Under Penetration
Before we get too comfortable with the power of the better handgun rounds, it is worth noting that penetration has been enough of an issue with heftier criminals in shootouts, even without ballistic vests, that it has been widely considered a problem.
And heavy winter clothing can even cause issues.
Glaser has had to come up with the silver glaser to deal with this and some agencies have revised calibre choices.
And while we both regard the medium power AK and the shotgun as being very adequate weapons, I have a friend who has been shot by kalashnikovs many times, and survived it all, usually killing the people firing at him.
It's true he has a round and muscular physique, an advantage in these cases, but it's not good to be too confident of these things.
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I found an interesting 45, by Para Ordnance.
6.5" by 4.5". 10 rounds.

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Some general specs on ammo-
* Caliber: 9mm Luger Plus P
* Bullet Wt.: 115gr Self-Defense DPX
* Velocity: 1250fps
* Energy: 399ftlbs
* Test Barrel Length: 4.0 Inches
* Caliber: 45 Auto Plus P
* Bullet Wt.: 185gr Self-Defense DPX
* Velocity: 1075fps
* Energy: 475ftlbs
* Test Barrel Length: 5.0 Inches
* Caliber: 7.62X39
* Bullet Wt.: 123gr Self-Defense DPX
* Velocity: 2300fps
* Energy: 1445ftlbs
* Test Barrel Length: 16.0 Inches
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* Caliber: 9mm Luger Plus P
* Bullet Wt.: 90gr COR®BON Self-Defense JHP
* Velocity: 1350fps
* Energy: 466ftlbs
* Test Barrel Length: 4.0 Inches
* Caliber: 45 Auto Plus P
* Bullet Wt.: 165gr COR®BON Self-Defense JHP
* Velocity: 1250fps
* Energy: 573ftlbs
* Test Barrel Length: 5.0 Inches
* Caliber: 45 Auto Plus P
* Bullet Wt.: 200gr COR®BON Self-Defense JHP
* Velocity: 1050fps
* Energy: 490ftlbs
* Test Barrel Length: 5.0 Inches
* Caliber: 7.62x39
* Bullet Wt.: 125gr COR®BON Self-Defense JHP
* Velocity: 2400fps
* Energy: 1600ftlbs
* Test Barrel Length: 18.5 Inches
* Caliber: 7.62x39
* Bullet Wt.: 150gr COR®BON Hunter JSP
* Velocity: 2300fps
* Energy: 1762ftlbs
* Test Barrel Length: 20.0 Inches
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Glaser Safety Blue
* Caliber: 7.62x39
* Bullet Wt.: 130gr
* Velocity: 2300fps
* Energy: 1527ftlbs
* Test Barrel Length: 18.5 Inches
* Caliber: 45 Auto Plus P
* Bullet Wt.: 145gr
* Velocity: 1350fps
* Energy: 587ftlbs
* Test Barrel Length: 5.0 Inches