momentum kills, well momentum has not killed a single cat as it is a mathematical construct only. But knowing about momentum p=mv can kill people and cats, some think twice the speed, twice the momentum, twice the impact... (not that any physicist think so, but still). nothing measurable correspond directly to the standard momentum (de Broglie related)...it is a derivative, that can be replaced by better stuff.

on the other hand kinetic energy, Compton momentum and also kinetic mass (yes a new useful concept) are all linked to v^2 (when v<<c), so yes if aware of any of them one will not double the speed. And yes we can measure such effects directly: Crash!

momentum came out from the time it was big confusion around if kinetic energy and moving masses where related to v^2 or v...it is a math artifact, that survived. If you take a cow and multiply it with the speed of light and divide it by 2/3 of a banana, then this construct is not a cow, even if it embedded has information about the cow. Sure it is useful to model the cow, if u don't know better. Better is to observe a cow and model it directly!