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I understand that in nuclear reactions such as fusion or fission, it is known that energy released due to the mass defect to obey the conservation of energy.

However, researching online; I fount that the conservation of momentum is also conserved! What is the reason, even though total mass is not conserved?

My reasoning, is that since there are no external forces, conservation of momentum is always true. But since there is a mass defect, then mv inital ≠ mv final.

How do I reconcile these two contradictions?

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It is not the case that the mass is not conserved. Granted, the sum of the rest masses of the nuclei after the fission, or the fusion, is less than the sum of the rest masses before. But the total mass, including the mass in the form of energy (according to Einstein relationship E=mc^2), is conserved. Mass in the form of energy also carries momentum. Therefore there is no contradiction.

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It is common to deal with several conservation laws at the same time in atomic and nuclear reactions, so it is not a one-or-the-other proposition.

In the case of fission, the problem can be analyzed by assuming that the fissioning nucleus is at rest in the reference frame of the lab before it splits. Then when the fission occurs, the momenta of the fission fragments sums to zero- and the masses you use to figure this out are the masses of the fragments after fission.

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