2004 v6 Rumble/Shake on startup
 

I have a 2004 v6 auto, 113,000 miles that I bought a couple of weeks ago. When starting up and slightly when turning off, for only a second or so after, there is a rumble/rattle which seems like it is coming from under my feet. It goes away after no more than 1.5 seconds. The rumble can only be heard inside the car. Can't tell if its actually rattling but I think I can feel something under my feet. I took the car in for this problem and a few other small things, car has an exhaust leak in the manifold and needs new tie rod ends. Mechanic said he doesn't know where the rumble/rattle is coming from. He checked the motor mounts, and transmission mounts and said they were fine. Also checked to see if the starter or anything else was loose, everything was tight. I was wondering if it could also be because of the exhaust leak? I don't know a ton about cars but I am learning quickly. Any help would be appreciated. Thanks!

Do you notice if it also happens when you stomp the gas while driving? Our generation of Mustangs seem to have a common issue with the stock exhaust manifold cracking. This can cause some ticking noises or vibration depending on how bad the problem is. Have an exhaust shop check it out.