Physicists often use the Dirac delta function to model this kind of “sharp pulse”; in this context, it would be the limit in which an infinite force acts over zero time. Of course, saying it that way makes it pretty ill-defined, but assuming you’ve got your formal mathematical definition squared away you can use it in many context like you would any other function.
As expressed above, the problem is not solvable. If you apply a constant force over an unknown distance or time, you can’t know the final velocity. And it’s even more impossible if you try to express the motive force as a WHUMP.
There it is.