Verified answer

This is not possible. Under General Relativity, as velocity increases, the rules of Velocity addition starts becoming blurred. If you were to take two space ships traveling at 0.9c (c is the Speed of Light), their speed relative to some third observer between them will only register as 0.995c.

Likewise, with your question, you can look at it with the formula:

u= (v+w) / (1 + [v*w/c^2]);

Where v and w are the magnitudes velocities of the protons, measured relative to the frame of reference of a third observer.

The velocity of Light will always be your limiting factor; as nothing containing information can travel faster than c.

Picture it like this:

car A travelling this way -> car B travelling this way <- they are both travelling at 100mph. When they colide it will be as if only one car that is travelling at almost 200mph colides into a brick wall.