This is the page to accumulate the information for the shuttle design. There will be links to part models, specification sheets, design sketches, and more.
A sketch of the shuttle design shows the various bits that need to get designed.
A video describing the design requirements (hopefully) makes the details in the sketch clearer.
Since the Vex 393 motor has several internal gearing options, we can use a 1:1 gear ratio to drive the bevel gears.
This will leave the output travel of the shuttle as about: .083 inches per turn * (50/80/120) turns per minute = 4.15/6.64/9.96 inches per minute = .069/.111/.166 inches per second. This might be a little slow, but it will give a reasonable proof-of-concept.
Parts and CAD files
48t, 24p Spur gear stock [CAD-zipped]
54t, 24p Spur gear stock [CAD-zipped]