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For example, a spaceship traveling very slowly from Earth to Moon stops somewhere halfway through (by "stops" I am presuming: "during a short time interval, his speed relative to both Earth and Moon is many magnitudes smaller than c").

Yet at this point, the spaceship (including the solar system) is moving at nearly light speed compared to a distant galaxy. So spaceship speed can, indeed, be considered "large enough" in at least some inertial frames.

According to Special relativity:

If a body's speed is a significant fraction of the speed of light, it is necessary to use relativistic mechanics to calculate its kinetic energy.

Also, as an object moves closer to the speed of light, it takes more and more energy to speed it up. But the spaceship above is "moving close to the speed of light" for a certain frame, even though it's stopped from Earth's reference frame. So, why would it take more or less energy to speed it up?

So what is its kinetic energy then?

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    $\begingroup$ Kinetic energy, like velocity, is not absolute and must be calculated in a specific reference frame. Ah, the joys of special relativity. $\endgroup$ – Rococo Jun 2 '17 at 18:15
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    $\begingroup$ @Rococo: that's already the case in Galilean/Newtonian physics $\endgroup$ – Christoph Jun 2 '17 at 18:21
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In special relativity the kinetic energy of a particle of mass $m$ moving with velocity $v$ is given by $$ K=(\gamma - 1) m c^2 $$ where $c$ is the speed of light and $\gamma=(1-(v/c)^2)^{-1/2}$. As $v$ is a frame dependent quantity and $m$ and $c$ are frame invariant, it follows that $K$ is frame dependent. For example in a frame comoving with the particle we have $v=0$ and so $\gamma=1$ and so $K=0$.

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