The change from .50 as the most common cartridge to .30 was due to the change from black powder to modern smokeless powder. The later burns faster and with more energy so you get a lot more energy in a smaller bullet. And the smaller caliber gives less air resistance so it goes further and more accuratly. So with the same length of barrel you end up with the same amount of energy in a .30 smokeless cartridge as in a .50 black powder cartridge but are able to shoot much further. This is why a lot of rifles were designed around very similar .30/8mm rifle cartridges.
Because normal firearms are meant to kill, (and in WW2, kill a man).
So, you need a round that has enough mass and kinetic energy that is can drop a man and make him stop fighting no matter wear it hits him. You also need a round that is effective at longer ranges.
But you also don’t want to have a round that is overkill, unless it’s for a specific gun that is expected to shoot through cover and light armor, you don’t want all your soldiers walking around carrying big ads heavy rounds, because they don’t need them. And the heavier the round, the fewer rounds each man can carry.
You also want you guns to be using the same caliber, because then keeping your Troops supplied is way easier since you don’t have to keep track of what kind of bullet everyone is using when everyone is just using the same.
So, Around 7.62 (or .30-06) was a round that just got commonly selected by everyone. For one rounds of that caliber were already common, so you could use the same factories you already have, and it gave you what you needed, around that wasn’t too heavy, but also had enough stopping power to be effective.
Just to name a couple or few reasons…
Ballistics are very good for the 7.62/.30-cal’s with holding accuracy, trajectory, and lethality at ranges that were common in WW1, WW2, and Korea. The 7.92/8mm & .303/7.7-cals were very close.
Anything larger was usually of more recoil with less accuracy down range. Anything smaller, (6.5’s say), were less lethal at longer ranges.
Edit: Should note, for those unaware, this is for rifles and smaller machine guns.
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