I have some problem to intuitively understand why the kinetic energy grows quadratically with the velocity (at least in non-relativistic case).
Assume the following experiment: we launch an unmanned rocket ship from an asteroid, let it accelerate time $T$ in a direction and then time $2T$ in the backwards direction; so, in time $2T$ it has velocity zero and in $3T$, it should be in the same position as in time $T$, just with opposite velocity. Then we let it fly freely until it hits the asteroid and let's assume that all kinetic energy transforms to heat.
Repeat the same experiment with $T$ replaced by $\lambda T$; the final velocity will be $\lambda$ times bigger and we used $\lambda$-times so much fuel for the accelerations. But this can hardly be converted to $\lambda^2$-times the energy.
Is the solution hidden in the fact that the fuel itself has a non-neglectable weight?