My ruling on this would be that there is no acceleration. Last peasant just drops it on ground. Thus making it pointless setup. That seems most consistent way to me.
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Really, the problem is the very selective application of real-world physics and game physics and then trying to very selectively obtain a particular outcome. If we want to play real-world physics, well, we all know the reasons why this isn't going to work in the real world. If you want to play D&D physics, then yeah, sure, the rod arrives at the other end of the line in one turn but with no more or less velocity than it started with, because "velocity" isn't even a concept in D&D. There is only "damage" in the D&D world, and there are no rules that state that handing off an item to the next person changes its "damage" any. Railgun a Greatsword from one end of the line to the other and a Greatsword still does 2d6 damage.
It does successfully demonstrate that the D&D rule set is a just a complete and utter failure as a Grand Unified Field Theory of Physics. I join the rest of the nerd world in shock and dismay at this outcome and encourage them to try harder next edition. If they'd just listen to my feedback and import the Standard Model this would all go away.
There's plenty of other ways to munchkin the rules to obtain absurd damage even completely within the ruleset. Fortunately. Or unfortunately. The reader may decide for themselves.
When a character falls, he suffers 1d6 points of damage for every 10 feet fallen, to a maximum of 20d6 (which for game purposes can be considered terminal velocity). This method is simple and it provides all the realism necessary in the game, It is not a scientific calculation of the rate of acceleration, exact terminal velocity, mass, impact energy, etc., of the falling body.
So accelerating the object (increasing its damage) up to some arbitrary cap sounds reasonable. Perhaps limited to twenty times.