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Assuing that our current understanding that the speed of light can't be matched or breached is correct, if you were travelling at the fastest possible speed in a space ship, say 99.99% of light speed, what would then happen if you were to fire a rocket out from the front of the ship?

Would it fire out normally from the ships prospective, in that case is it going faster than the speed of light for someone looking in from another angle? Or would the energy propelling the rocket forward be converted into mass, in which case the rocket would remain where it is and get heiver?

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Yes it fires out normally from the ships perspective, because from ships perspective, the ship is at rest.

No it does not go faster than light from the point of view of another observer, because velocities are not added the same way as in classical mechanics, but are added by relativistic velocity-addition formula.

In classical mechanics, when ship fires rocket with speed say $u=0.2c$ from its perspective, than from the perspective of observer who sees the ship moving with speed $0.99c$ the rocket has speed $u'=1.19c.$ Not so in special relativity. Here the formula says: $$ u'=\frac{0.99c+0.2c}{1+\frac{0.99c\times0.2c}{c^2}}\approx 0.993c $$

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There seem to be some fundamental mis-understandings.

Put yourself inside the rocket. However fast you go with respect to any chosen other body in the universe, your rocket would work exactly the same way, irrespective of your (relative) volicity.

What you will notice: the faster you go wrt this other body, the longer your rocket will need to use fuel (thus you need more energy) to become faster (but you will notice this only when approaching approx. speed of light relative velocity).

At no point energy is converted into mass (and I don't want to dwell here on mass of binding energy of propellant as that's besides the point of this question) - even when the relativitistic mass of your rocket increases when views from the other body you choose to compare your velocity with

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