Unfortunately this question doesn't really fit easily into here or askscience, but I've got my fingers crossed since many on Reddit tend to be multidisciplinary and generally inquisitive.
Late 18th and early 19th century navies made use of both carronade and long guns.
I know the general idea was a much heavier but slower short range cannon ball for the carronade, while the long gun made use of lighter but faster shot.
My question is this: Was there any difference in force/damage inflicted, in say, Joules?
A fast light object can impart just as much energy as a slow and heavy one. And with the Carronade being much less accurate I'm wondering what the upside was.
Adapted from a previous answer
Carronades were designed to have a "smashing" effect due to their relatively large balls and small charges; the sacrifice in range was considered worth it for the enormous holes they could punch in enemy ships (or for the effect they could have as antipersonnel weapons loaded with grape or musket balls). They were often called "smashers" for that exact reason.
To skip back a couple hundred years: The problem the English fleet had around the time of the Spanish Armada was that the guns of the time had relatively long muzzles and largish charges, leading to high-velocity shot punching through ships without creating much damage to them. Many of the casualties in the period I'm interested in were caused by splinters breaking off from the ship and causing lacerations among the crew. To achieve that, English and later British gun founders accepted lower muzzle velocities and somewhat larger caliber of balls, especially when iron cannon became more widely available in England. Even though iron guns are arguably worse than bronze, they were still well-liked by shipbuilders.
The most common type of carronade -- the kind you'll see if you visit HMS Victory -- fired a 68-pound ball. The largest long guns on the Victory were bored to take a 32-pound ball, and themselves weighed 56 long hundredweight, or in modern terms, something just north of 6200 pounds. (There's a reason the long guns on Victory were replaced by fiberglass -- they were slowly breaking its back.)
An important factor to consider about the use of corronades is the ease of reload for a shorter gun. Rate of fire can be greatly increased on a shorter piece as each action required for reloading will take less time and effort. This benefit is mostly important at shorter ranges such as broadside duels where time is a factor and in which a lower velocity shot still has enough energy to do significant damage to the enemy ship and crew. In addition, with a shorter barrel, windage and variation in bore size and shape can be more easily controlled and thus the powder used per shot can be reduced even for a larger bore and shot size. The gun would also be significantly lighter for a given shot size. The accuracy and reduction in downrange energy for the shot is the main driver for ships using corronades along with traditional long guns as the complication of having to store multiple sizes of shot is greatly outweighed by the increased versatility of the ships under combat.