Sitting on my desk is the day-calendar for "Common Errors in English Usage" and today's error discusses the phrase the carrot on the stick. Only, that phrase is incorrect. Yes, I've been using it too. The correct phrase is the carrot or the stick and refers not to a carrot dangling from a stick, but rather to rewarding someone with a carrot or beating them with a stick. Likely a mule, but this proposition is believed to also be effective in negotiations with wives, children, co-workers, and Jehova's Witnesses as well.
I made that last part up.
So, take it from the day-calendar beside me, if you've always envisioned a little man riding a donkey with a carrot dangling from a stick by a string held out in front of the donkey as a lure to get him to move... YOU WERE WRONG. The correct picture in your mind should be a man standing in front of the donkey with a carrot in one hand and a stick in the other and a little cartoon-caption balloon saying, "What's it going to be pal... the veggie or the lumber?"
Oh, and by the way, I accidentally ordered... umm... a lot of these day calendars so if you would like on, please do let me know and we can work something out. You can check them out here. I'll cut you a good deal.