With due respect to Mr. Weatherby, I've not seen any reference other than Layne Simpson's article on the history of the .30-378 exceeding 6,000 fps, and it would be very interesting, for historical and technical reasons, to get information on how it was measured (what equipment was involved) and what the magic propellant was.
Consider that a 30-378 loaded with a light 100 grain bullet. At 6,000 fps that bullet would carry about 8,000 ft-lbs of muzzle energy. About the same as a 460 Weatherby Magnum getting 2450 fps with a 600 grain bullet, which Hodgdon claims it can do with about 115 grains of H4831 and a 26" tube (though QuickLOAD thinks the pressure would still be too high; Hodgdon has no pressure measurement for their 460 Weatherby loads, so this is one of those "work-up-to-very-carefully "published loads). So that makes the 6,000 fps 100 grain bullet seem at least feasible—until we consider the powder mass.
In the 460 Weatherby, the velocity is low enough, and the expansion ratio large enough (about 9:1) that virtually all the powder burns in the barrel, and so, on average, with half that gas at the breech standing still, and half chasing the bullet, we have 57.5 grains of powder mass at 2450 fps the moment before the bullet base clears the muzzle, which takes only 767 ft-lbs of energy, or less than 10% of the total provided by the powder. If we try to drive that same powder mass to 6,000 fps to chase the 100 grain bullet, it takes 4,600 ft-lbs of energy just to move half the powder, a much bigger drain on the powder energy budget. It gives us 8,000 + 4,600 ft-lbs or 12,600 ft-bs that has to be extracted from the powder and put into forward-moving translational kinetic energy. The original 115 grains is no longer adequate. Now we need at least 115×12,600/8,000, or 188 grains of the same powder. But that doesn't cut it either, because even if you could assume all the powder could burn behind the light bullet in the smaller .30-378 expansion ratio (4.3:1, which it won't; ballistic efficiency will decrease), we now have a bigger mass and now even more energy is needed to move the powder and so the total charge increases further. Indeed, if the same 69.6 ft-lbs of muzzle energy per grain of powder could be extracted in the 30-378 as with the 460 Weatherby, it would take a 270 grain charge of powder to reach 6,000 fps with the 100 grain bullet. Basically, twice as much powder as it takes to get to 5,000 fps under the same assumptions. To fit that 270 grains into a 30-378 case, you would need powder density about the same as the solid density of powder with no space between the grains.
So, maybe a miracle propellant was delivered to Mr. Weatherby, or maybe there was a velocity instrumentation error. But 6,000 fps is really stretching credibility. I read long ago that the military had experimented with 30 and 50 caliber bullets in necked-down 20 mm cases in super heavy fixed guns, but ran into a limit for nitrocellulose-based powders that was below the 6,000 fps mark, though close to it (the number 5,700 fps comes to mind, but my memory should not be trusted on this). Powder mass and ballistic efficiency decreasing with degree of overbore case capacity sets limits to velocity that are pretty hard to overcome if nitrocellulose-based propellants are used, even with nitroglycerin added to increase total energy density. Some other, higher energy density propellant may be another matter. If we assume the veracity of the 6,000 fps number, I can only venture the opinion that it wasn't likely done with a nitrocellulose-based powder. I don't know what it would be, though. At any rate, I hope this gives the OP a better idea of the nature of the problem. It's the exponentially growing quantity of powder required, even if bores and throats are to be sacrificed.
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