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Since mass increases with speed (apparently), while being accelerated to 99.999 percent of the speed of light, what does one of those little protons weigh when they collide?

Okay- I brought up the question since many documentaries on physics refer to the fact that as speed increases, the mass(weight, whatever) increases

as you approach the speed of light. That it is "impossible" to travel at the speed of light because the the mass becomes so great, etc... a sort of a "catch-22" situation develops. Maybe a simpler comparison would be- "IF a proton weighed 1oz at it's static point, what would it weigh at the speeds developed in the LHC?

Apparently, a photon has no mass,

As for the 3 quark construction, I am aware that they are part of the fundamental building blocks of particles like protons, etc... Does your question mean that some Q's have no mass, thus negating some of the effects?

weight is a relation to mass, mass is a relation (equivalence) to 'energy' per Einstein.

At a relativistic speed the mass grows,

sometimes called relativistic mass, or just mass energy. Lightarrow speaks about the 'energy' of a particle, to differ it from its 'proper mass'

which is the mass defined from matter as Earth being in a uniform motion, in a ideally flat vacuum without any other gravitational influences. The proper mass of Earth won't change if we double Earths, and its solar systems, speed. As long as it is in a uniform motion that is, but its relativistic mass or 'mass energy' must, as far as I get it.The headache for me is that you can give this solar system any speed you like. But locally you won't see any difference in energy, as your light bulb suddenly emitting gamma radiation, through that increased speed. So the relativistic mass, or 'mass energy' created through a different relative (uniform) motion is in fact undetectable from any local measurement, as proven by your scale as you weight yourself. The way to prove its existence is through collisions, or locally defined accelerating.

That's what the LHC does, although I'm not sure if they allow uniform motion at the collision, or if they just keeps accelerating the particles, all the way up to a collision. The idea is that even though locally undetectable in a uniform motion it will exist as a cost for you, accelerating, forcing you in the end to expend a infinite 'energy' to reach the speed of light in a vacuum.