Chadwick's discovery of the neutron involved consideration
of collisions between alpha particles and protons and between alpha
particles and nitrogen nuclei. 
Here we prove the formula stated on the page discussing
the discovery of the neutron. 
Before collision, we assume that the proton (or
nitrogen nucleus) has negligible speed (ie we assume it's
stationary) 






After collision, both particles are moving 





If there are no external forces acting, momentum is conserved,
therefore 


and, if the collision is elastic, K.E. is conserved, therefore, 


Eliminating v from these two equations gives something like this 


which, after a few minutes messing about gives 


QED which stands for "quite easily done"... (don't believe
what anybody else tells you about these letters...) 