How is it that when momentum is constant, energy always stays a constant.
For Example :- if $P = 0$ kinetic energy will also be 0.
But in a explosion (that momentum is conserved) where the object was still before the explosion the momentum still should be 0. By using $E_k=\frac{P^2}{2m}$ energy also return as 0 Jules.
If the $E_k=\frac{mV^2}{2}$ is used the kinetic energy comes as $E_k>0$ since kinetic energy is a scalar.
How is that?
The total energy consists of several contributions. Kinetic energy is one of these, but there is also potential energy. Clearly if you stand still at the top of the stairs your kinetic energy is zero, but your potential energy is not and is converted into kinetic energy when you fall down. Chemical energy is more complicated. A chemical explosion is driven by the fact that the initial components have smaller binding energy than the final ones. The difference is converted into kinetic energy and heat. Binding energy of a component consists of electronic kinetic and potential energy.
However from the body of your post it is clear that you are asking something else, namely if there is a relation between the total momentum of a system and its total energy. This is only the case for a system without internal degrees of freedom. If it has internal degrees of freedom, these can have energy without contributing to the total momentum, neglecting relativistic effects via the rest mass.
This may not always be true. Consider a piece of nickel wire at above its Curie temperature, where it is non-magnetic. The electron spins are randomly oriented and there is no net momentum to the electron gas as a whole - just as many electrons are moving in one direction as in any other. Below the Curie temperature the latter is still true, but now the electron gas has adjusted itself to a lower energy state since the spins have now oriented themselves.
Edit: I did some further investigation on this and it seems that this may not be a good example. The ferromagnetic transition is of course second order, so there is no latent heat release during the transition. For iron, it seems that most popular idea for a while was that the electrons reduce their coulomb repulsion at the expense of increased kinetic energy at its ferromagnetic transition. But a paper came out in 2014 that challenges that and concludes it is the opposite.
https://journals.aps.org/prb/abstract/10.1103/PhysRevB.90.125102
So there is some controversy in this basic system! The point is I think, though, that the total energy of the ferromagnetic system is a constant and there is just an exchange, one way or the other, between the potential and kinetic energy of the system. I don't think this point is usually covered at all in most basic textbook discussions of ferromagnetism. At least, I can't find it in any of mine.
It can be said that an isolated system which does not emit or receive radiation must conserve energy, but you must consider all of the different forms that energy can take.
