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In a news article today the reporter claims that modern improvements in "ammunition design" have made long rife barrels unnecessary for assault weapons. Here is the paragraph:

The 20-inch barrel isn’t needed anymore.

One of the biggest differences between the M4 and the M16A4 is the barrel length: 14.5 inches versus 20 inches, respectively. The M16A4’s longer barrel allowed for a higher muzzle velocity and a longer effective range; conversely the shorter M4 barrel limited its performance in both categories. But improvements in ammunition design have enabled the M4 to close the gap with its longer forefather.

The Marine Corps adopted a new 62-grain, 5.56×45 mm Special Operations Science and Technology round. The SOST round is designed to perform out of barrels as short as 10.5 inches, so the M4 has no difficulty shooting out to the extreme end of effective range, negating the advantage of the long M16A4 barrel.

What improvements in ammunition design are they referring to? Why would a bullet be equally accurate coming out of short barrel?

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The main reason that the SOST round performs as well as the legacy M855 in terms of range and accuracy (all while being shot from a shorter barrel) is because it was designed to be fired from the shorter barrel from the get-go. This mostly comes down to adjusting the propellant geometry and burn rate so that it all burns up before the bullet exits the gun. The M855 propellant was designed to burn up within the 20 inches of bullet travel it got in the M16. Firing it from a 14 or 10 inch barrel puts it way outside its original design envelope, so it's no wonder that it's lacking in performance when fired from the M14 or the SCAR. There's also a trade-off between barrel wear and muzzle velocity; you can use a hotter propellant that will give you more velocity, at the expense of shorter barrel lifetime.

This does raise the question as to why anyone used a 20-inch barrel in the first place, if the same performance can be had from a 10.5-inch barrel. The reason is that the legacy M855 round has much better armor-piercing capabilities than the SOST. That article leaves this part out, and in doing so doesn't make a fair comparison between the two rounds. The M855 was designed during the cold war, where it would likely be used against enemies wearing some body armor or against light-armored vehicles. For defeating those kinds of targets, you need velocity. The M855 fired from an M16 has a few hundred feet per second over the SOST fired from the SCAR as far as I can tell.

Some of the performance improvements of ammo is of course due to advancements in technology, but very often it's just the result of re-optimizing a few system components to adjust to changing requirements. I think that's the main reason for the perceived improvement of the SOST over the M855.

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Looks like this report, and the links therein, are seriously lacking in technical info. I did a quick lmgtfy and found some info on the munition in question at the firearm blog. Quoting:

SOCom developed the new rounds for use with the Special Operations Force Combat Assault Rifle, or SCAR, which needed a more accurate bullet because its short barrel, at 13.8 inches, is less than an inch shorter than the M4 carbine’s. Using an open-tip match round design common with some sniper ammunition, SOST rounds are designed to be “barrier blind,” meaning they stay on target better than existing M855 rounds after penetrating windshields, car doors and other objects.

I'd guess this round would still have greater range & accuracy when launched from a long-barrel rifle, but it exceeds the Marine Corps' specs for range and accuracy. There appear to be a large number of non-round reasons why the M16 is being replaced, including maneuverability, burst mode operation, and so on.

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