Can kinetic energy be transferred between two objects even if they are not in contact?
This question is better explained with a thought experiment. It is inspired by this answer, stating that the amount of work done depends on the inertial frame.
Consider a one-dimensional space with two spheres $1$ and $2$ of equal mass and a grenade $g$ directly in the middle between them. In a general case, the spheres have kinetic energy $E_1$ and $E_2$, though a restriction to non-relativistic velocities is imposed. The mass of the grenade shall be negligible such that its kinetic energy is zero, however it has internal energy stored $E_{g,i}$ which will be released on the spheres upon detonation. Thus, the system has an initial state $i$ and a final state $f$, before and after detonation of the grenade, respectively. The grenade releases the entirety of its internal energy at the detonation $E_{g,f} = 0$.
Compare the following two inertial frames:
1. All objects are initially at rest $v_{1,i} = v_{2,i} = v_{g,i} = 0$, thus $E_{1,i} = E_{2,i} = 0$. 512
After detonation and according to symmetry, both spheres receive an equal amount of energy from the grenade $E_{1,f} = E_{2,f} = E_{g,i}/2$ and accordingly $v_{1,f} = -v_{2,f}$.
2. All objects move with a constant velocity $v_{1,i} = v_{2,i} = v_{g,i} = \tilde{v}$. For simplicity's sake, the energy released by the grenade is just the right amount, such that $v_{2,f} = 0$ and $E_{2,f} = 0$. This is the case precisely when $E_{1,i} = E_{2,i} = E_{g,i}/2$. Now, what is the energy of the other sphere after detonation $E_{1,f}$? It is the only object in the final state of this system that carries energy. Thus, by conservation of energy $E_{1,f} = E_{1,i} + E_{2,i} + E_{g,i} = 2E_{g,i}$.
For me this result is problematic. In the second case, sphere $1$ seems to consume both the initial kinetic energy from sphere $2$ and the full internal energy of the grenade, though the spheres appear to receive equal and opposing forces from the explosion in the rest frame of case one.
Plugging in some numbers, if the grenade contains 4J of energy, then in case one each of the spheres receives 2J from the explosion, while in case two 6J of work were done on sphere $1$ since it ends up with 8J of kinetic energy.
I think there is something fundamentally wrong about the way I imagine energy transfer. Can you spot a mistake or help me out? 512