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Why would'nt this work?
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Because the stick isn't infinitely rigid. If you push it at one end the other end doesn't immediately start moving. The time it takes, I think, is equal to the speed of sound inside that material. Ultimately the forces that bind atoms together and allow them to interact are limited by the speed of light.
I ran this by an engineer and they said the same thing
Huh....so we may fail to achieve faster than light (FTL) travel but we could probably manage faster than stick (FTS) travel
Easily. I imagine that most spacecraft are already traveling faster than the speed of stick. It's likely only a few thousand meters per second