I haven't found the right search terms for this question, so if it has been answered, references would be welcome.
Suppose we start from experimental station in deep space (interstellar space if need be; ie: very low gravitational interaction), where we launch a photon powered rocket. That is, it sends photons out one end at the speed of light to produce thrust with minimal reaction mass loss, say with a laser. This produces a constant force and a constant acceleration (ignoring any minuscule reduction of the mass of the rocket due to radiated energy).
With constant thrust, the velocity V of the rocket will be a linear function of time, and the accumulating delta momentum mV will also make sense.
But it would seem that the kinetic energy of the rocket (1/2 at^2), relative to the station, is also accumulating at the square of time. The rate of accumulation of kinetic energy (energy per second, or power) is increasing linearly with time.
The same laser "engine" seems to get more and more "efficient" (producing more accumulating kinetic energy with passage of time). The second ten minutes of operation appear to generate 3 times the kinetic energy that the first ten minutes of operation produced (and 5 times as much during the third ten minutes, etc).
This seeming paradox is so obvious that it must be a stock issue resolved long ago, but I'd appreciate being brought up to speed. (No pun intended)
(Obviously there are going to be still other effects if the rocket were to approach relativistic speeds. I'm looking more for the solution within classical mechanics – like hundreds or thousands of meters per second).
Best Answer
I apologize in advance, but I'm going to make the problem look worse first so that I can explain it:
Imagine that instead of you measuring the velocity from the station where the rocket is launched - let's call it station A - you measure instead from station B, which is moving away from station A at some high speed. And imagine that station A launches two identical rockets, one toward you and one away from you.
Since we're operating under the well-tested current theory of special relativity, we understand that we can either say "B is moving away from A" or with equal correctness "A is moving away from B" since we can declare either one as "stationary" for purposes of measuring velocities with respect to our chosen coordinate system. So let's start over and say that Station A is moving at speed away from Station B and launches two rockets.
Of course, before they launch from A, both rockets will have a certain kinetic energy based on their motion, and that energy will be equal for the two identical rockets. When we launch them, however, that begins to change: one of them is starting to move away from us even faster, while the other begins moving away from us slower. After some time, in fact, one will be moving away from us at twice the speed it originally had - that is, four times the kinetic energy - while at the same time the other is moving away from us at zero relative velocity: no kinetic energy whatsoever! Now we're really mixed up, because both rockets did the exact same thing but one ended up with a very high kinetic energy and one with no energy whatsoever.
The issue I hope to highlight with that example is that kinetic energy is frame-dependent and not an intrinsic energy for an object. That's why you can safely ride along in a car moving at 100km/h along a road but not be safely struck by a car moving at 100km/h along a road; if you travel with the car, it has zero kinetic energy in your frame, but if you're standing on the road it has a very high kinetic energy in your frame. Kinetic energy is relative.
On another note, I said before that the two rockets are doing "the same thing," but in any given single frame of reference that's not quite true. Obviously, in my example above, from the frame of station B one rocket is losing kinetic energy and the other is gaining kinetic energy, for example. But that's only considering the rockets themselves: those are being propelled by launching photons the other direction, and those photons are expelled with a certain energy themselves. If you were to measure the energy of the photons themselves, (attaching a small mirror to the back of the rocket headed toward you so that you can see the "exhaust") you would notice that over time the exhaust of the rocket moving away from you seems less energetic and the exhaust of the rocket moving toward you seems more energetic. That is to say, photons launched toward you are less energetic than photons launched away from you... which was the same conclusion we came to with the rockets themselves launching from Station A, and again is a consequence of the relativity of kinetic energy. But this time we can more directly say that one is "red-shifted" and the other "blue-shifted."
EDIT: It may also be simpler to imagine the solution if you use a normal reaction drive with massive exhaust. The exhaust is low mass but leaves at high speed; the rocket is higher mass and thus gains less velocity from the reaction. The kinetic energy increases by the same amount in opposite directions for each body of mass (taking all the exhaust as a "body"), otherwise we're getting energy for free from somewhere. Also, since massive exhaust has subliminal expulsion speed, you can have a case where the rocket is accelerating away from you but the exhaust is also moving away from you, which isn't possible with photon drives. But in all cases, photons or not, the total energy of the system is conserved: measured from an inertial coordinate system, the rocket "gains" the same amount of energy that the exhaust "loses," and the measured exchange rate does increase quadratically.