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Relativistic momentum is gamma.m0.v, which diverges from the classical value as speed rises. A proton at 0.95c has gamma = 3.203, so its momentum is more than three times the Newtonian estimate. Accelerator design depends on this: beyond a few percent of light speed the classical formula badly underestimates the magnetic field needed to bend a beam.
Relativistic momentum
p = gamma m0 v
Energy-momentum relation
E^2 = (pc)^2 + (m0 c^2)^2
Above about 0.1c the error passes 0.5%, and by 0.5c momentum is understated by 15%. Below 0.01c the difference is negligible.
Yes - p = E/c. The energy-momentum relation still holds with the rest mass term set to zero.