Imagine there are two particles: A,B; of which, we observe A to be stationary, while B is moving towards A at velocity v. A,B has same mass; and they are charged both positive.

B has a potential wrt A. After some small time t, B's potential increases by an amount X; so it's kinetic energy has been reduced by the same amount X. This amounts to B losing some velocity y, although it still has some velocity towards A. The kinetic to potential conversion means that B has it's net mechanical energy unchanged till now.

Now, in that time, A has gained the amount y of velocity, away from B (conservation of momentum). That means A has gained some kinetic energy K. Of course, from the beginning time where A was stationary, and till now, A has continuously faced slow potential increase: each increases instantaneously dropped to add to it's kinetic energy: so that now we have this kinetic energy K. Let, the potential energy of A wrt B was, at the beginning, P. Then the final potential of A ultimately is not lower than P. Then the total mechanical energy of A has actually increased till now.

So counting now, we have that the sum of energy of A,B has actually increased: in violation of conservation of energy.

I know, it's the little increases of potential of A, which I can't explain: why they are gained: before contributing to the kinetic energy of A. It seems that these potential increases suddenly popped out from nothing.

I have seen an example of ball falling from high to ground: there potential is exactly what is converting to kinetic energy. Everything was fine, until I thought about the circumstance I described.

Imagine there are two particles: A,B; of which, we observe A to be stationary, while B is moving towards A at velocity v. A,B has same mass; and they are charged both positive.

B has a potential wrt A. After some small time t, B's potential increases by an amount X; so its kinetic energy has been reduced by the same amount X. This amounts to B losing some velocity y, although it still has some velocity towards A. The kinetic to potential conversion means that B has its net mechanical energy unchanged till now.

Now, in that time, A has gained the amount y of velocity, away from B (conservation of momentum). That means A has gained some kinetic energy K. Of course, from the beginning time where A was stationary, and till now, A has continuously faced slow potential increase: each increases instantaneously dropped to add to its kinetic energy: so that now we have this kinetic energy K. Let, the potential energy of A wrt B was, at the beginning, P. Then the final potential of A ultimately is not lower than P. Then the total mechanical energy of A has actually increased till now.

So counting now, we have that the sum of energy of A,B has actually increased: in violation of conservation of energy.

I know, it's the little increases of potential of A, which I can't explain: why they are gained: before contributing to the kinetic energy of A. It seems that these potential increases suddenly popped out from nothing.

I have seen an example of ball falling from high to ground: there potential is exactly what is converting to kinetic energy. Everything was fine, until I thought about the circumstance I described.

Imagine there are two particles: A,B; of which, we observe A to be stationary, while B is moving towards A at velocity v. A,B has same mass; and they are charged both positive.

B has a potential wrt A. After some small time t, B's potential increases by an amount X; so it's kinetic energy has been reduced by the same amount X. This amounts to B losing some velocity y, although it still has some velocity towards A. The kinetic to potential conversion means that B has it's net mechanical energy unchanged till now.

Now, in that time, A has gained the amount y of velocity, away from B (conservation of momentum). That means A has gained some kinetic energy K. Of course, from the beginning time where A was stationary, and till now, A has continuously faced slow potential increase: each increases instantaneously dropped to add to it's kinetic energy: so that now we have this kinetic energy K. Let, the potential energy of A wrt B was, at the beginning, P. Then the final potential of A ultimately is not lower than P. Then the total mechanical energy of A has actually increased till now.

So counting now, we have that the sum of energy of A,B has actually increased: in violation of conservation of energy.

Imagine there are two particles: A,B; of which, we observe A to be stationary, while B is moving towards A at velocity v. A,B has same mass; and they are charged both positive.

B has a potential wrt A. After some small time t, B's potential increases by an amount X; so it's kinetic energy has been reduced by the same amount X. This amounts to B losing some velocity y, although it still has some velocity towards A. The kinetic to potential conversion means that B has it's net mechanical energy unchanged till now.

Now, in that time, A has gained the amount y of velocity, away from B (conservation of momentum). That means A has gained some kinetic energy K. Of course, from the beginning time where A was stationary, and till now, A has continuously faced slow potential increase: each increases instantaneously dropped to add to it's kinetic energy: so that now we have this kinetic energy K. Let, the potential energy of A wrt B was, at the beginning, P. Then the final potential of A ultimately is not lower than P. Then the total mechanical energy of A has actually increased till now.

So counting now, we have that the sum of energy of A,B has actually increased: in violation of conservation of energy.

I know, it's the little increases of potential of A, which I can't explain: why they are gained: before contributing to the kinetic energy of A. It seems that these potential increases suddenly popped out from nothing.

I have seen an example of ball falling from high to ground: there potential is exactly what is converting to kinetic energy. Everything was fine, until I thought about the circumstance I described.