Does a trebuchet with wheels actually shoot further?
The answer is that they sometimes increase range. In the case of a fixed-counterweight machine they do, and if the FCW is large it may be unstable without them. With a hinged-counterweight treb they may or may not, depending on the proportions of the arm and counterweight. Generally with this type of treb it doesn't help much. But wheels can help with transport.
This is a fallacy based on the Nova TV show on trebuchets filmed at Loch Ness. Two types of trebuchet were constructed. One was the hinged counterweight type that does not need wheels. This is the type more commonly used in the Middle Ages, and is the better of the two. All the historic trebuchet recreations in Europe were built without wheels. See the one at Warwick Castle as an example. The other has the weight fixed rigidly to the arm. Because of that, when the weight drops it's forced to follow a curved path, which first pushes the axle back and then pulls it forward. This has two effects. It can make the frame rock dangerously back and forth and it wastes energy doing it. The wheels allow the weight to fall almost vertically (the frame is pushed back, then pulled forward) so it's both safer and allows more energy to go into the throw. Because the fixed counterweight treb was built by an American team it got more coverage and they showed why the wheels were needed. They also showed why the French-built hinged CW machine didn't need the wheels but gave it a lot less time.
A okorder /wgbh/nova/lostempires... Many other sources below. Wingman