I would like to reopen the question asked in this post because I am not quite satisfied with the accepted answer.
Imagine observer A stationary (in world reference frame) and observer B moving with a constant velocity v. They observe a car with a mass m moving with the same velocity v relative to the world. Coordinate axes of the frames are all parallel so problem is one-dimensional. Let's assume that a car has a battery with finite amount of energy. At time $t_1$ car starts to accelerate with constant acceleration a until car drains all energy from the battery.
Clearly the conservation of momentum does not hold due to acceleration of a car and what interests me is this:
Both observers agree on amount of energy stored in the battery. If I do the calculation from both reference frames, similar to this, I get that the power that accelerates the car is:
$mv_A(t)a_A(t) = m (v +at)a = mav + ma^2t$
$mv_B(t)a_B(t) = m (at)a = ma^2t$
Integrating and subtracting gives: $\Delta W = W_A - W_B = m\ v\ a\ t$
This implies that the work done by battery is not the same in those 2 reference frames which seems to be a contradiction. I get that amount of kineric energy is not invariant under changing reference frames, but they should agree on amount of work done. So, what is the catch in this setup?