My car makes a thud like noise what is coming from the back underneath area anytime I back up or pull forward initially. It doesn’t make it after I start driving? Just with the initial acceleration and stopping. There is nothing inside the trunk rolling around but that’s kind of what it sounds like.
You need to have someone look at the wheel to see if it moves forward/backward any when you are hearing the noise, could be a trailing arm bushing bad or if you had alignment lately may be the rearward lower adjustable control arm (toe bar) is loose…
You probably need to lift the vehicle and start trying to move the wheel and suspension links/parts around to find any movement, also check the exhaust and basically everything that is not the body…
Or pay a professional shop to check it out for you…
Hiring a pro shop per the above post to do a proper diagnosis is the most certain and safest solution. But if you are diy’er inclined, you could ask a helper to walk along outside the car to narrow it down to one side or the other; then remove that wheel and give it a shake. One possible theory, there may be a small pebble stuck in the wheel and it rattles around when you first start moving. When you start moving at a faster pace the centrifugal force of the rotating wheel holds the pebble in place.