How long does force apply? If you were to throw a ball into the air while standing on earth, it will take some time to grow from zero relative velocity to it's maximum height before the velocity starts to increase again as its falls. Can we know how long this time period of acceleration from zero to maximum is and why?
 A: Summarizing some of the comments (since they have a tendency to disappear over time):
The force lasts as long as you apply it. Force is not a property of material - momentum is. As you know, momentum of a particle initially at rest will be $F\Delta t$ after you apply a force $F$ for a period of time $\Delta t$. More generally, the change in momentum
$$\Delta(mv) = \int F\cdot dt$$
When no further forces act on the particle, it will then continue with that same momentum "forever". 
If you throw an object, it is accelerating while you apply force on it - while you throw it; after you release it, it will continue with constant velocity unless other forces (drag, gravity) act on it.
A: I see in the comments section this question became something else entirely, but I'll try to sum up all points in my answer -- forgive me if I repeat what others said.
First, why should the proton not accelerate forever? It will surely accelerate as long as the force acts upon it. I think what you mean is akin to the Zeno's Paradoxes, in which basically it is studied whether something existing for an infinitesimal amount of time could have a finite effect, that is something that is measurable (in your case, a force existing for almost no time generating a finite, countable speed). The answer is basically no, in order to create a finite speed, the force must act for some time bigger than zero. And I would add, stop making that fruitless thought experiment right now :)
Now, in a universe with nothing but the proton, a net force on the proton could hardly be generated because Newton's Third Law establishes that an opposite force would act upon another object. If there's no other object, then any pair of forces would be internal to the proton, cancelling each other if we analyze the entire proton. But that's nitpicking, and I understand your thought experiment anyway.
About the ball scenario you described in the comments: understand that the net force on the ball will be a discontinuous function. Its integrals, however, which are speed and distance, will be continuous. I'll try to make a graph:

In this graph your hand accelerated the ball upwards for 1.5 seconds, the released it into the air. See that although the force (red line) went abruptly from 2.2 to -2.4, the speed (green line) did not change abruptly (although it began to change abruptly). Even more, the ball height (blue line) is perfectly smooth.
SUMMING UP: even though the FORCE changes abruptly, a finite (nonzero) amount of time is needed for that change to be reflected in speed and displacement. Contrary to Zeno's Paradox, things that exist for "infinitesimal" periods of time have no "real" implications.
