Question
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?

a) $v_{He}' = 1.53 \times 10^7 \textrm{ m/s}$

$v_{Au}' = 3.13 \times 10^5 \textrm{ m/s, } 29.8^\circ \textrm{ below positive x-axis}$

b) $7.84 \times 10^{-13} \textrm{ J}$

Note: As noted in a comment below, I made an error in the solution video where, in equation 4 at around 10:26, I should have canceled the factor 2 in the term $2v_{\textrm{He}\textrm{He'}}$, in which case the linear term of the quadratic equation should be half of what's shown in the video. Correcting this error results in the final answers shown above which, apart from $v_\textrm{Au'}$ are very close to what's shown in the video. This solution has been flagged for correction later.

Second note: A common question is "what happens to the $\cos{180 - \theta_\textrm{He}}$?" Since $\theta_\textrm{He} = 120^\circ$, this term becomes $\cos{60} = \dfrac{1}{2}$ (my forgetting of which resulted in the first note...)

Solution Video

# OpenStax College Physics Solution, Chapter 8, Problem 49 (Problems & Exercises) (17:52) Rating

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## Calculator Screenshots      Video Transcript

Submitted by justavet136 on Fri, 10/04/2019 - 21:07

What happened to the cos(180-theta He in after you added 1b^2 and 2b^2?

Submitted by GraceWe712 on Fri, 01/17/2020 - 15:37

What happened to the cos(theta HE) ?????

Submitted by 18310174593marshal on Mon, 08/03/2020 - 23:31

Hi I wonder, at 10:26, equation 4, shouldn't the 2 v He v Au be just v He v Au, since cos (180 - theta hellium) is 1/2 cancels the 2 in front of the term?

Submitted by ShaunDychko on Wed, 08/05/2020 - 14:57

Astute observation! I have made corrections to the final answers after accounting for this error. Please see the notes in the final answer for more details, and thanks again for noticing this.