This problem has nagged me for a long time, and yes; it IS rocket science!

Assume a rocket with mass m in free vacuum space switches the on its engine for time t at a set throttle level.  The fuel-burn is constant, so force developed is constant, and hence the rocket's velocity will rise in a linear fashion.

(For simplicity I have ignored the change in mass due to fuel useage; let's assume it's an ion drive)

So after time t the velocity is v, and the kinetic energy of the rocket is (1/2) * m * v * v

The rocket now coasts along for an indeterminate time using no fuel, and maintaining its kinetic energy.

The engine is now switched on again to repeat exactly the acceleration as above for the same time and with the same fuel useage.  It's velocity is now 2v (isn't it; it has, after all, carried out two bursts of identical acceleration)

The problem now is that the rocket's kinetic energy is (1/2 * m * (2v) *(2v)), and this is four times the kinetic energy it had after the first fuel-burn.

So the question is this: how can the rocket gain more (double, in fact) energy kinetically than has been used in the two fuel-burn sessions.

Clearly there is an error, and onecan easily demonstrate by other means how the books do actually balance.  The problem I find is in identifying the error in the above reasoning.