[pzhon]

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Relativistic mass is sort of a misnomer. Its mass doesn't actually increase.

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Yes, it does, according to some conventions. No object can move faster than c, but its momentum is unbounded. If you want, you can think of the additional mass as the mass of the kinetic energy.

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Remember we have the conservation of mass and energy laws. If it were to increase, then these laws would be violated.

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No, they wouldn't.

If you ignore relativistic mass, and only use rest-mass, you make it much harder to describe conservation of momentum, and the fact that light has momentum despite its zero rest-mass.