Voltage required to accelerate a proton into another one

1. The problem statement, all variables and given/known data

A proton is accelerated through a voltage ∆V with parallel plates towards another stationary proton that is "far away" outside of the proton gun. To get fusion, we want them to get close enough to touch at the surfaces. Find the voltage accross the plates that would accelerate the proton enough so that it will collide with the stationary proton.

2. Relevant equations

U = kqQ/r^2
U = 1/2*QV
Conservation of energy; KE = 1/2 mv^2

3. The attempt at a solution

Letting KE at the "front" of the proton gun be 0 and U at the end be 0, the voltage will accelerate the proton to KE=U=1/2*QV (similarly, let KE=0 at the point where the protons touch). The next step is where I got stuck; since touching the other stationary proton would require r to be zero, wouldn’t the proton have to overcome an infinitely large potential energy to come in contact with the other one?

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