How much energy would be needed to accelerate a large spaceship to half the speed of light is?

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At lower speeds, the kinetic energy is 1/2 mv2 (1/2 times mass times speed squared), but since we are approaching the speed of light, the more accurate formula is (note: I am using root() to indicate the square root):

mc2(1 / root(1 - (v/c)2) - 1), which in this case is (mass in kilograms) x (300,000,000 m/s)2 x (1 / (root(1 - 1/4)) - 1) = mass x 9x1016 x (1 / (root(3/4)) - 1) = mass x 9x1016 x (1 / 0.866 - 1) = mass x 9x1016 x (1.155 - 1) = mass x 9x1016 x 0.155 = 1.395 x 1016 (Joule for every kilogram).


Note 1: This is the kinetic energy content of the spaceship, and the minimum amount of energy required to accelerate it to that speed; if any energy is wasted, the total needed would of course be more.


Note 2: The standard (non-relativistic) formula is 1/2 x mass times (150,000,000 m/s)2 = 1.125 x 1016. This is somewhat lower than if you use the correct relativistic formula. The more you approach the speed of light, the larger will be the discrepancy between the relativistic formula and the classic formula. At less than 1% of the speed of light or so, the discrepancy is so small that it isn't worthwhile to use the more accurate (and more complicated) relativistic formula.
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