[mrerick]

Barrel length is a balance between utility, power and accuracy. That is why you see different lengths used.

The longer the barrel length, the higher the potential velocity a given bullet weight and powder load will achieve. It's simply because the high pressure expanding against the back of the bullet spends more time accelerating it.

At the time that Lugers were engineered, instrumentation for measuring the high pressure and velocities, including terminal velocity and energy dissipation, were pretty crude and empirical in nature.

Starting on page 230, the red Sturgess book discusses development of the Luger Carbine, including the history of pistol carbines to that point and issues relating to barrel length. Luger prototype pistols had a range of barrel lengths, presumably for use in testing. A table on page 315-316 shows a large number of experimental lengths.

Ultimately, selection of the barrel length was a compromise between utility / carrying convenience and ballistic performance. Each military commission had it's own priorities. Eventually things settled down to standards.