energy of the system increasing after acceleration (one reference frame) The system consists of small mass $m$, large mass $M$, and chemical potential energy $E_{chem}$, as shown in below image, the reference frame is following the small mass throughout.

The total energy of the system is:
$$ΣE=(m+M)\times{c^2}+E_{chem}$$
The small mass uses the chemical potential energy to accelerate, reaching a velocity of $v$ where $v$ is calculated as such:
$$v=\sqrt{\frac{2\times{E_{chem}}}{m}}$$

After acceleration, the larger mass will appear to be moving and thus have a kinetic energy: $$E_{K_M}=\frac{M\times{v^2}}{2}$$
This makes a new total energy of:
$$ΣE=(m+M)\times{c^2}+\frac{M\times{v^2}}{2}$$
Consequently the energy change from before to after the acceleration is:
$$ΔΣE=E_{chem}-\frac{M\times{v^2}}{2}$$
Since when $E_{chem}$ and $m$ are constant, $v$ will also be constant, meaning that $M$ can take on any value, and make the net energy change non-zero. Which of the many physical laws that was ignored would come to rescue?
 A: Here is the question: what do we mean by "kinetic energy"? I assert that we mean "the amount of energy or 'work' I would need to make this object come to rest in my frame of reference".
Consider a ball with mass $m$ in free space 
$${\huge\circ}$$
In it's inertial frame of reference it's velocity is zero so its kinetic energy is also zero. But all inertial frames are equivalent, or at least I cannot say that one is more correct than another so let's consider one which is moving at a velocity $v$ relative to the ball. In this frame the ball is moving and has kinetic energy $E_k = mv^2$.
$${\huge\circ} \\ \rightarrow v$$
Thus kinetic energy is a relative quantity that depends on your frame of reference.
A: When you say the small mass will have a certain velocity, v, after converting the chemical energy - what frame of reference is that in?  Not in this a frame of reference following the small mass because in that frame of reference the small mass's velocity must always be zero.  For that step you're using what I'll call the page frame of reference - presumably and inertial frame.  If you then switch to considering things from a frame of reference fixed on the small mass you've switched to an accelerating, non-inertial, frame of reference.  That's what's messing up your conservation of energy.
