POTW #01: The Hexagonal Space Station

$\textbf{Note:}$ All parts can be solved using standard techniques from the high school physics Olympiad syllabus.

$\textbf{Optional Story:}$ Rey is an engineer evaluating different structural prototypes for a modular space station. To test the integrity of her candidate hexagonal models, she triggers a sudden impulse on an outer segment. She must quickly calculate exactly how these unanchored structures will recoil and fold. Therefore, she models them as follows:

$\textbf{Problem:}$ Consider a system of identical uniform rigid rods, each of mass $m$ and length $2L$, initially at rest on a frictionless horizontal plane. The rods are connected at their endpoints by frictionless hinges. In each scenario below, an impulse is applied strictly perpendicular to a rod at its center of mass.

Bonus (Not graded): For a chain of $n$ such regular hexagons, what is the initial velocity $v_n$ of the outermost rod of the $n$-th hexagon in terms of $u$?