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The conservation of energy tells us that energy before the collision equals energy the comes out of the system after the collision. In the case of a stationary target nuclei and a neutron beam we have as our initial energy

Rest mass (E - mc2) of the nuclei and the particle in the beam.

Kinetic energy of the nuclei and the neutron projectile.

So the initial energy looks like

KEn + mn*c2 + KEnuc + mnuc*c2

We note that the initial KE of the stationary nucleus is zero and omit this term.

KEn + mn*c2 + mnuc*c2

Assuming neutron absorption (i.e. only 1 particle) all of the energies after the collision are described by

KEnuc_f + mnic_f*c2

Use the principle of conservation of energy to set the two equal to one another.

KEn + mn*c2 + mnuc*c2 = KEnuc_f + mnic_f*c2

Now solve for the final KE of the combined nuclei, KEnuc_f

KEnuc_f = (KEn + mn*c2 + mnuc*c2) - (mnic_f*c2)

This is the recoil energy.

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