[Tristan Lall]

That's really fast...I don't think those Velmex slides are right for you. They're good for moving heavy stuff slowly and precisely.

It looks like you need to move light stuff quickly and precisely (if 47 W is the low end of acceptable, you won't be moving much mass at that speed). I can't think of any off-the-shelf parts that would work nicely for you.

Are you building a pick-and-place machine of some sort? The ones used to manufacture circuit boards can achieve approximately the sorts of speeds of operation and travel you're looking at—with relatively small loads—but they're not even remotely in your budget. (Maybe you can find pieces of one of those for sale at an industrial auction or something?)

For something that can't quite manage that sort of speed, but is at least reasonably close, check out an inkjet or dot matrix printer head. They have very accurate position control built right in, if you can figure out the control scheme.