On earth, catapult-based transportation has never really taken off. There are many valid reasons for this, most of which we will attempt to ignore completely for the sake of half-bakering.
The four reasons most pertinent to this discussion are as follows:
1: The gravity of the earth makes any
travel involving an upward component somewhat less efficient than others, and places troublesome limits on weight/distance calculations.
2: The earth has already been trod upon, with at least rugged foot paths leading to most areas.
3: unpredictable wind patterns make flight paths significantly unpredictable. Passengers tend to avoid transit on vehicles with unpredictable paths that cannot be internally controlled.
4: Extensive resources for fuel and existing means of non-projectile transportation are readily available.
On the moon, this situation is far different. The moon currently contains only a few outdated, and probably inoperable moon buggies for transportation at the moment. Pathways are poorly trod, and in widely separated areas. Each separate set of pathways meanders around what is now a pile of rubbish, useful only for spare parts or as a tourist attraction.
What pathways are there are of high historic value, and adding foot prints or tire tracks to the areas may be undesirable.
The moon has virtually no atmosphere, so the course of any trebuchet-launched transport would be affected only by the constant effects of local gravity fields, and the extremely faint force of solar winds, which can only affect the illuminated portion of the globe, and which can be predicted about 1/2 hour ahead of time more reliably than terrestrial winds.
Using large trebuchets to hurl passengers from point to point across the lunar globe, and then catching them in large nets, or complicated devices resembling arrestor cables may be a very efficient means of transportation. The pathways can be reliably established ahead of time, but roads or long rail lines do not need to be constructed. Also, there is less friction, so less energy would be wasted, and if the launching/catching complexes are constructed well, energy absorbed to catch one vehicle could be put into preparing another trebuchet for launching.