As part of his discovery of the neutron in 1932 , James Chadvick determined the mass of the newly identified particle by firing a beam of fast neutrons, all having the same speed, at two different targets and measuring the maximum recoil speeds of the target nuclei. The mum speeds arise when an elastic head-on collision occurs between a neutron and a stationary target nucleus. (a) Represent the masses and final speeds of the two target nuclei as $m_{1}, v_{1}, m_{2},$ and $v_{2}$ and assume Newtonian mechanics applics. Show that the neutron mass can becalculated from the cquation
$$
m_{n}=\frac{m_{1} v_{1}-m_{2} v_{2}}{v_{2}-v_{1}}
$$
(b) Chadwick directed a beam of neutrons (produced from a nuclear reaction) on parafin, which contains hydrogen. The maximum speed of the protons ejected was found to be $3.3 \times 10^{7} \mathrm{m} / \mathrm{s}$ . Because the velocity of the neutrons could not be determined directly, a second experiment was performed using neutrons from the same source and nitrogen nuclei as the target. The maximum recoil speed of the nitrogen nuclei was found to be $4.7 \times 10^{6} \mathrm{m} / \mathrm{s}$ . The masses of a proton and a nitrogen nucleus were taken as 1 $\mathrm{u}$ and 14 $\mathrm{u}$ , respectively. What was Chadwick's value for the neutron mass?