8.6 Exercise 8

Professional Application

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 $\left(^{4} \text{He}\right)$ from gold-197 nuclei $\left(^{\text{197}} \text{Au}\right)$. The energy of the incoming helium nucleus was $8.00 \times \text{10}^{- \text{13}} \text{J}$, and the masses of the helium and gold nuclei were $6.68 \times \text{10}^{- \text{27}} \text{ kg}$ and $3.29 \times \text{10}^{- \text{25}} \text{kg}$, respectively (note that their mass ratio is 4 to 197). (a) If a helium nucleus scatters to an angle of $\text{120}º$ 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?