Thanks for opting healthcaremagic.com to post your query and for giving us this opportunity to clarify your doubts.
Well, since its 24hrs now that your son doesnt have any complaints, he most probably just gulped it down. There is a possiblity that whenever children gulp down anything from their mouth so rapidly, it can enter into their air passage and cause breathing difficulty. That also depends on the size of the stick. Anyways, he should have had at least some symptoms of pain or breathlessness by now. Since he seems to be well, I dont think there is anything to worry. I would still advise you to keep a watch on him for any new signs of forceful vomiting
, breathing difficulty for at least a day. He should be absolutely well though.
I hope I have been successful in allaying all your anxieties and you have got the answer to your question. Take care and stay healthy.