Ernest Rutherford (the first New Zealander to be awarded the Nobel Prize in Chemistry) demonstrated that nuclei were very small and dense by scattering helium-4 nuclei $\left( ^4\textrm{He} \right)$ from gold-197 nuclei $\left( ^{197}\textrm{Au} \right)$. The energy of the incoming helium nucleus was $8.00 \times 10^{-13} \textrm{ J}$, and the masses of the helium and gold nuclei were $6.68 \times 10^{-27} \textrm{ kg}$ and $3.29 \times 10^{-25} \textrm{ kg}$, respectively (note that their mass ratio is 4 to 197). (a) If a helium nucleus scatters to an angle of $120^\circ$ during an elastic collision with a gold nucleus, calculate the helium nucleus’s final speed and the final velocity (magnitude and direction) of the gold nucleus.
(b) What is the final kinetic energy of the helium nucleus?