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Considering that speed increases weight and the proton is going at almost the speed of light, I would like to know how much a speeding proton would weigh in the LHC.

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you mean mass, not weight – Val Cool Aug 3 '15 at 8:14

Real quick, comparing 4 Tev to 1 Gev or 900 Mev, about 4,000 times as much as when it's sitting still.

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