If I was reallly motivated I could use some of my graphing tools at work and plug in velocity and barrel lengths to create a curve of best fit for acceleration. However. I firmly believe that the highest acceleration rate is almost instantainious.
I think to be most accurate you would eed to start the graph with no barrel. eg a bullet in a vise whacked with a hammer and use that as the 0" baseline. If you get that fps for a give round, then shoot that same same at 3 or 4 lengths I think you would get a pretty accurate measurement as to what the curve would look like.
I honestly don't believe that the actual barrel length plays a massive role in the acceleration rate and the final velocity of a bullet.
I have always believed that that barrel lenght provides the ability to stabilize the bullet during that acceleration.
For example if Bullet X with X amount of powder to burn can accelrate Bullet X for say 20 inches of barrel lenght before the preasure is no longer growing enough to accelerating the bullet than it does not good what so ever to have a barrel of 24 inches. In fact the bullet will leave slower in some instances.
Of course the more powder you have the greater the initial preasure and initial acceleration burst..ths why we have magnums. However there come a point lke I said where the shockwave as the bullet leaves the barrel will actually server to de-stabilize the spin rate of the bullet as it leaves the barrel, the barrel will flex too much, etc.. etc... In fact I would think if you really really overpowered a given round the round would tumble.
Now I think I am just saying what I already said.
I think the most basic thing to understand are.
1) Any powder that burns after the bullet leaves the barrel donothing but add a flame effect
2) A massive ammount of the actual acceleration that takes place does so at a 0" barrel length
3) If powder is not burning for the full duration of the bullet in the barrel the round is under powered(now weather or not a given barrel, breach or shoulder can handle the maximum amount of powder is not known to me, perhaps you could still accelerate effectivly a .223 with 2 pounds of powder.
4) The more powder you need the longer it takes to burn thus you would need a longer barrel to allow for that complete powder burn.
These are just my somewhat logical assumptions based on not much fact but it makes sense to me.
It helps to think of it in terns of extremes for example a 7mm win mag shot out of a 0 inch barrel, a 1 inch barrel and an 80 foot long barrel.
In the 0 inch barrel the bullet would still be going damn fast, but not spinning and the bullet would not fly very straight
the 1 inch barrel the bullet would atleast spin a bit but probably not be very stable in that it has only spun about 1/10th of a turn and would probably stop spinning at some point. The preasure would be increased and the bullet would go faster.
In the 80 foot long barrel the bullet would likely not leave the barrel and at some point in that 80 feet it wold stop accelerating and would start to decelerate. That would be the ideal barrel lenght in my eyes. How long that distance is would depend on the ammount of powder and the bullet characteristics(boat tail, bullet length, fit of the bullet in the barrel smoothness of the bullet and barrel, no different than a piston in a car engine)
Everything is atrade off right? How much force can your shoulder take, how much preasure can the barrel take without distorting too much to affect trajectory, What is the ideal spin rate of the bullet and how long does it take to accheve that spin rate with engough momentum to maintain that spin rate down range...
The way to really get an idea of this whole topic of velocity in relation to barrel length is to like I said cronograph a bullet with no barrel.
Than shoot the same setup through a 22 inch barrel a 20 inch barrel and an 18 inch barrel then that should be enough to get a good estimatation of the accleration curve.
My guess?
Ok here goes.
I am guessing that in a round that shoots at 3000fps out of a 22 inch barrel. I would think that the same round would go at least 2/3 that speed if shot with no barrel at all.
So that leaves 1000 feet per second of acceleration over 22 inches. So that would be an overal average acceleration rate of 45 fps per inch of barrel.
However I still think that the barrel provides greater efficiency down at teh breach end.
But I think my numbers are pretty close.....anyone feeling crazy with a vice and a crono..haha just kidding I would not recomend doing this it could be very dangerous. Better leave this one to myth busters.