It can look dumb, but I always had this question as a kid, what physical principles would prevent this?
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The speed of βpushβ is effectivly the speed of sound in a medium. So your shove would be the same as propagating a soundwave through whatever that rod is made of.
Veritassium covers this https://www.youtube.com/watch?v=EPsG8td7C5k&t=61s