The old timers sorted out how long a barrel needed to be and as powder got better the barrel got shorter. Looking at late guns and into the black powder cartridge rifles 30-32 inches became the normal length for full sized rifles. Carbines had short barrels but shot pistol size loads.
UM...., I'm uncertain about the validity of that assertion.....
Short barrels existed long long before the powder "got better". Long barrels existed where the powder was quite good, and persisted.
Short barrels are much easier for men who use them from horseback, especially in forest that are not old-growth such as those found in Germanic areas that spawned the very short Jaeger Rifle. At the same time those same Germanic rifle smiths already had longer barreled rifles for hunting on foot, and even longer barreled rifles for target shooting.
The American long rifle was a reapplication of certain rifle design principles already known in Germany. Longer sight plane and smaller caliber for men who dismount and hunt on foot for deer in North America..., gave highly efficient powder and ball combinations and excellent accuracy (compared to the short, huge caliber rifles of the common Germanic Jaeger and his rifle :wink: )
Carbines existed too, long before the powder was improved, and existed where the powder had no need for improvement. France had very short rifles and smoothbore carbines, and they had great black powder, as did the Germans, in the 18th century. The British with their poor powder had carbines for mounted troops and artillery as early as the mid 18th century, and during the F&I War.
As for pistol cartridges and carbines, that's a function of ammunition capacity for combat before needing a reload, coupled with recoil considerations (imho). While the Spencer (a cavalry carbine), Sharps (a cavalry carbine), and Burnside carbine (another cavalry arm) all shot cartridges with less powder than the rifles of their time.... the 1873 Springfield rifle shot the same cartridge whether full sized infantry or much smaller carbine. Though they did reduce the powder in the cartridge to lessen the carbine recoil for cavalry troops. :wink:
To answer the OP, with today's excellent black powder, and even "cleaner" burning substitute powders, the short barrel (36" or less) gives you ease of handling and transporting, while the longer barrels give you better sight radius so it makes it easier for your eye to make an accurate shot when the distances to the target get to around 100 yards, and less recoil. That's about all there is to it. The fact that you might have to use 95 grains in a .50 caliber, traditional, White Mountain Carbine to get the same velocity that somebody gets from a 42" barrel on a caplock .50 shooting 70 grains of powder, is a moot point where powder resupply is pretty easy compared to two centuries in the past.
LD