"can a slow billiard impart its energy to a faster billiard?"
/sci/ said no, with various arguments. but it turns out it can. pic related.
In this scenario (frictionless, elastic) where the collision tangent is more or less vertical, the slower ball exchanges its x momentum with the faster, and does not exchange any y momentum.
so the fast ball now has all of the system's x and y momentum

if you aren't just willing to accept being a retard, figure out what was wrong in your reasoning before