Actually, i hadn't thought about that, i am currently wondering whats the best way to cut down some 3.5mm screws (for the combo refit) but since you mention it....
Whilst reversing at warp speed in a airplane on a conveyor belt which is turning the opposite way to the forward motion of the airplane, if i were to attempt cut some 3.5mm screws with an electrically powered cutting tool, would the tool work?
It might be worth noting that the bottom belt of a conveyor belt moves in the opposite direction to the top belt. Thus although the top belt may be going in the opposite direction to the plane, the bottom belt is not.
No one has mentioned the Lorentz contraction. This effect would be particularly apparant at warp speed, which is greater than the speed of light. At such speed, the contraction would result in negative lengths, somewhat akin, IMO, to a person disappearing up his own bottom orifice.
The Enterprise doesn't travel faster than light. Warp drive works by warping space, hence the name. Roughly speaking, you shrink space in front of the ship and stretch it behind. The measured speed as seen from outside the warp field is faster than light but, inside, you are travelling at sub-light speed through compressed space.
The idea of warping space has some serious science behind it. Have a look here:
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