$[0/3$ Points$]$

OSCOLPHYS$28\mathrm{.}6.$WA$\mathrm{.}055.$

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 $({?}^{4}He)$ from gold$-197({?}^{197}Au).$ See the figure below. The energy of the incoming helium nucleus was $8\mathrm{.}20{10}^{-13}J,$ and the masses of the helium and gold nuclei were $6\mathrm{.}68{10}^{-27}$ and $3\mathrm{.}29{10}^{-25}kg,$ respectively $($note that their mass ratio is $4$ to $197).$ If a helium nucleus scatters to an angle of $120$ during an elastic collision with a gold nucleus, calculate the helium nucleus's final speed.

$15668249\mathrm{.}989$

Put the origin of the coordinate system at the site of the stationary gold nucleus and the $+x$ direction in the direction in which the incoming helium nucleus travels. Write a statement of conservation of energy, momentum in the $x$ direction, and momentum in the $y$ direction to obtain three equations and three unknowns. See if you can solve the three equations simultaneously for the three unknowns. $\frac{m}{s}$

Calculate the final velocity $($magnitude and direction$)$ of the gold nucleus. $($Assume the positive $x$ direction is the direction in which the helium nucleus is initially traveling, and that it scatters $120$ clockwise from the $+x-$axis.$)$

magnitude

$\frac{m}{s}$

direction $($counterclockwise from the $+x-$axis$)$