10 week old coughing so hard he vomits?

My 10 week old son has a head cold. He saw the dr a few days ago who said that we would see a lot of drainage in the next week or so but didnt say anything about vomitting. Whenever my son coughs extremely hard, I feel the need to pat his back and he throws up almost evertime (about 2-3 times a day) It is not spit up. He is throwing up a lot of food (milk). I am scared to not pat his back because I do not want him to choke on the flem. We think the flem is gagging him and making him throw up. he only vomits during big coughing fits. Any suggestions. Is the vommitting worrisome to anyone else?