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\begin{center}
  {\bf University of Kentucky, Physics 404G \\
    Final Exam, 2020-11-27 }
\end{center}

Instructions: This take-home exam is due on Sunday 2020-11-29 at 11:59 pm.  You may not consult with any other source besides the course textbooks and the lectures and notes posted on the course website.  In-person or online help is strictly prohibited, including searches on the internet or online homework solution sites.  Show intermediate work for partial credit.

 
{\bf 1. Pretzel-bungee-jumper}

[10 pts] {\bf a)} Write the Lagrangian in circular coordinates $(\rho,\phi)$ of a person of mass $m$ hanging under the weight $mg$ of gravity from a bungee cord, approximated as a spring of natual length $\ell$ and spring constant $k$.

[10 pts] {\bf b)} Use Lagrange's equations to calculate the equations of motion $\ddot\rho(\rho,\phi,\dot\rho,\dot\phi)$ and $\ddot\phi(\rho,\phi,\dot\rho,\dot\phi)$, and identify the Coriolis and centrifugal terms.

[\phantom{1}5 pts] {\bf c)} Integrate the equations of motion numerically and plot the trajectory $(x(t),y(t))$ for the first two minutes, given $m$=100~kg, $\ell$=10~m, $k$=100~N/m, $\rho_{0}$=1~m, $\phi_{0}$=10$^{\circ}$, $\dot\rho_{0}$=1~m/s, $\dot\phi_{0}$=0.

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  {EXF_fa20_fig1}
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{\bf 2. Hard scattering with recoil}--A projectile ball of mass $m_{a}$ and radius $r_{a}$ is thrown with velocity $\bh{x} v_{a}$ and impact parameter $s$, bouncing elastically off of a stationary ball ($v_{b}=0$) of mass $m_{b}$ and radius $r_{b}$, which recoils backward from the collision.

[\phantom{1}5 pts] {\bf a)} Calculate the initial velocity $\bh{x} v'_{a}$ of the projectile in the center of mass reference frame, in which the total momentum equals zero.

[\phantom{1}5 pts] {\bf b)} Calculate the final velocity $(v'_{x},v'_{y})$ of the projectile in the center of mass frame.  Using conservation of energy and momentum and assuming an impulse perpendicular to the surface of impact, the ball `reflects' from the plane of incidence, as it does in Taylor Example 14.5.  Note that the plane of incidence is stationary in the center of mass frame.

[\phantom{1}5 pts] {\bf c)} Calculate the final velocity $(v_{x},v_{y})$ and scattering angle $\theta$ of the projectile in the laboratory frame.

[\phantom{1}5 pts] {\bf d)} Calculate the scattering cross section $d\sigma/d\Omega$ for this interaction.

[\phantom{1}5 pts] {\bf e)} What is the total cross section?

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