We actually don't know how fast a T. Rex could move. Various palentologists have suggested speeds ranging from as low as 11 mph up to 30 mph. Horner thinks they were slow movers, based on both the Rex's leg proportions and the fact that if it toppled over while running (not unlikely given its high center of gravity) it would break a bunch of bones and probably die.
The running speed of T. Rex has been mathematically modeled and, based on the assumption that Rex weighed 6-8 tons, it could not have physically moved faster than 18 mph. Other palentologists argue, based on the femur length, that T. Rex had a walking speed of 12-15 mph. The debate now is on the Rex's weight, as if it was (as some palentologists think) more like a 4-ton animal than a 6- or 8-ton animal, it could have moved a bit faster. Current estimates of its running speed range from about 12 mph to 25 mph.
And, it wasn't really very agile, either. Its center of gravity was six feet off the ground, its arms puny, its weight considerable by any estimation, and its legs bulky. This was not an animal that jumped or made quick turns because again, (1) its body wasn't capable of doing that, and (2) if it fell over it would quite possibly die due to shattered bones and internal injuries.
The shape of T. Rex's teeth don't go to show anything other than that it was a carnivore, and that is not in any dispute. Sharp teeth are useful to meat-eaters, whether the meat is alive or dead. That's why we have steak-knives instead of butter-knives for cutting meat.
Big head size is good for eating great big mouthfuls of flesh, also generally helpful if you are a large carnivore no matter whether you scavenge or you hunt. Again, that shows nothing other than that Rex was a carnivore, and we knew that.
Given that T. Rex had both traits that would assist a predator (e.g., binocular vision) and traits that would assist a scavenger (e.g., a terrific sense of smell), it seems logical to think that T. Rex was probably an opportunistic feeder -- it hunted if it had to or if prey was readily available, and scavenged when it found a fairly fresh carcass. But we can't tell for sure, based only on its bones, exactly how it fed.
