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the rocket will never reach lightspeed, or even close.
the faster something goes the more massive it becomes. the more massive it becomes the harder it is to increase the velocity of that massive object ever higher. way before reaching light speed, the energy needed to accelerate the object will be more than is available by any means. it would esswentially take infinite energy to get an object with any mass (even a proton) to go light speed.
hell, you have seen how much energy is used to just send a beam of protons in a particle accellerator at 3/4 the speed of light! Enough to power small cities! and that is just the mass of a few particles!
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