\documentclass{article}
\usepackage{exam3}
\usepackage[colorlinks=true]{hyperref}
\usepackage{enumerate}
\paper{PHYS 2210}               % <- do not include FC, FT etc

\title{Homework 2}
\time{Start of class on Thurs. Jan 26}
\semester{Spring}
\year{2012}                     % <- default is the current year
%\campus{City}
\note{ Show your work and explain what you are doing. (Correct answers, for which we cannot follow the work, are worth no credit). As always, unless otherwise stated, it's 4 points per lettered sub-section. }
\begin{document}
\begin{questions}

 \question In the great blizzard of '89, a rancher  had to drop hay bales from an airplane to feed her cattle. Suppose the plane flew horizontally at a steady 120 km/hr, and dropped the bales from 60 m above the flat range. (Neglect air resistance in the first two parts of this question)
 \begin{enumerate}[(a)]
 \item If she wanted the bales of hay to land 20 m behind the cattle (so as to not hit them!), where should she push the bales out of the plane? (Clearly define a coordinate system so we understand your answer) 
 \item  What is the largest time error she could make while pushing the bales out of the plane, to ensure not hitting the cattle? 
 \item If we did NOT neglect air resistance for the dropping hay, describe qualitatively (no detailed calculations, please!) what would change about your answers to both previous questions. 
 \end{enumerate}
 
 
 \question A soccer player kicks the ball  with a velocity $v_0$ at an  angle $45^{\circ}$ (to reach maximum range). She wants to kick it in such a way that it  barely passes  on top of two opposite team players of height $h$.

\begin{enumerate}[(a)]
 \item Show that if the separation between the two opponents is $d\leq\frac{v_0}{g}\sqrt{v_0^2-? gh}$, the soccer player succeeds.  (What is the numerical value of the ? in this formula?) 
 \item Use a computational tool of your choice (Mathematica or Python) to plot $d$ as a function of $v_0$,  setting $h=2.0\  m$. To do this, write the equation as a function rather than ``hard-coding'' the equation into the plot function.\\Additional help on writing functions in Mathematica is available here: \url{http://youtu.be/1A4f91yMVhA}
  
 %{\it Hint: If you want to plot a function $f[x]$ vs $x$ with $x_0<x<x_1$ with Mathematica you can use the instruction ${\rm Plot}[f[x],\{x,x_0,x_1\}]$. Sqrt[] is the syntax for square roots. Note  that Mathematica can't plot a function with symbolic coefficients (so you have to plug in the values for both  $g$ and $h$) Important: when you type a command into Mathematica, you must hit Shift-return, rather than just return, to have it evaluate what you typed!!}.

     \item Give a physical explanation of any major features in your plot.
     \item What are some benefits of writing functions to plot over ``hard-coding'' plots?
 \end{enumerate}
 

 
\question (8 pts) Trajectory of a particle. A particle  moves in a two-dimensional orbit defined by

\begin{eqnarray}
x(t)=\rho_0[1  + \cos (\omega t)]\\
y(t)=\rho_0[2 + \sin(\omega t)]
\end{eqnarray}

\begin{enumerate}[(a)]

\item Sketch the trajectory.  Find the velocity and acceleration (as vectors, and also their magnitudes),  and draw the corresponding velocity  and acceleration vectors along various points of your trajectory.  Discuss the results physically - can you relate your finding to what you know from previous courses? Finally: what would you have to change if you want the motion to go the other way around?
\item Plot the trajectory using your favorite computational tool. For this plot, set $\rho_0 = 1$ and $\omega = 2 \pi$. This is a called a ``parametric plot''.
\item Prove (in general, not just for the above situation) that if velocity, $\vec{v}(t)$, of any particle has constant magnitude, then its  acceleration is  orthogonal to $\vec{v}(t)$. Is this  result valid/relevant for the trajectory discussed in part a? 
     
     {\it Hint  There's a nice trick here -  consider the time derivative of $|\vec{v}(t)|^2 = \vec{v}\cdot\vec{v}$}. 
        \end{enumerate}
     {\it Note: What you have proven in part c is quite general, and very useful. It explains why, e.g., $d\vec{\hat r}/dt$ must point in the $\vec{\hat \phi}$ direction in polar coordinates. Do you see why?}         
      
 

\question   Let's generalize  the analysis you did in class (for the  motion of a particle in  polar coordinates) to {\it spherical} coordinates.
The three unit  vectors: $\hat r$, $\hat \theta$ and $\hat \phi$  which describe spherical coordinates  can be  written as:
\begin{eqnarray}
\hat{r}&=& \sin\theta \cos \phi \hat {i}+\sin\theta \sin \phi \hat {j} +\cos\theta {\hat k},\label{coo1}\\
\hat{\theta}&=& \cos\theta \cos \phi \hat {i}+\cos\theta \sin \phi \hat {j} -\sin\theta {\hat k},\\
\hat{\phi}&=& -\sin  \phi \hat {i}+\cos\phi{\hat j}.\label{coo}
\end{eqnarray}

\begin{enumerate}
\item First, investigate that the definitions given in Eqs.(\ref{coo1}-\ref{coo}) make sense.  Define in your own words what orthonormal vectors are, and then check to see if these vectors $\hat{r}$, $\hat{\theta}$ and $\hat{\phi}$ are indeed orthonormal. Then, convince yourself (and the grader) that e.g. at least the x-component of $\hat{r}$ is correct, using a simple geometric picture. (Taylor's Fig 4.16, or Boas Fig 4.5 should help) 

\item If we constrain the particle to move with  $\phi=0$,  state in simple words what this means in terms of the particle motion. Sketch the three spherical unit vectors at some point $\phi=0$ and $r=R$ for some particular (nonzero) angle $\theta$ of your choice.

 \item Show that the velocity of any particle in spherical coordinates is given by:\begin{eqnarray} \vec{v}=\dot r \hat{r}+ r\dot \theta \hat{\theta}+r \sin\theta\dot \phi \hat{\phi}\label{vee}\end{eqnarray} 
 
 \item Let's switch to  {\it cylindrical} polar coordinates. Find expressions for the cylindrical unit vectors ($\hat\rho, \hat\phi,$ and $\hat z$) in terms of Cartesian coordinates ($\hat i, \hat j, \hat k$) Then, take a derivative with respect to time to find $d\hat\rho/dt$ in cylindrical coordinates. 
%\item Use  the previous part to compute the angular momentum $\vec{L}=\vec{r}\times \vec{p}$ in terms of $\hat{r}$, $\hat{\theta}$ and $\hat{\phi}$, and then show that $L_z=mr^2\dot{\phi}\sin^2\theta$.

% {{\bf Hint!}  Just think about these three unit vectors -   can you very quickly and intuitively argue what the cross product of $\hat{r}$ is with each of the spherical unit vectors without doing any calculations at all? Just be careful of signs.}

\end{enumerate}
%{\it Extra credit} (up to 4 bonus points, but won't count off if you don't do it)
 %If a particle is restricted to move on the surface of a sphere, $r(t)=R$,   and  if angular momentum (in particular $L_z$) is conserved (meaning, $L_z$ doesn't change with time) {\it and assuming the particle starts its motion somewhere on the z axis}, what can you say about $\dot\phi$ at later times?

\question Romeo and Juliet are in love, but in our version of this story, the characters are a little different than tradition holds. Let R(t) = Romeo's feelings for Juliet at time t. Large positive values means deep love, large negative values means strong hatred. Similarly, Juliet's feelings for Romeo will be characterized by J(t). Now let's consider a ``model for the lovers" governed by the general linear coupled differential equations,
\begin{eqnarray}
\frac{dR(t)}{dt}= a R(t) + b J(t)\\
\frac{dJ(t)}{dt}= c R(t) + d  J(t)
\end{eqnarray}
%is referred to as the Lotka-Volterra equation and is supposed to represent
%the evolution of the populations of a predator and its prey as a function of time,
%$A, B, C, D$ are positive constants.(12 pts: 4 each)

If a$>$0 and b=0, I might describe Romeo as ``in love with his love". His emotional state has nothing to do with Juliet's feelings at all; he just falls more in love the more in love he is\dots a very narcissistic individual! 
\begin{enumerate}
\item Let's assume none of the parameters are zero. Consider {\it all four} possible permutations of signs of a and b, and for each, describe the emotional character of Romeo. (For instance, one of the combinations might be called a ``emotionally distant jerk", which one? But maybe you have a better or totally different description/interpretation of this same combination!) 

\item Suppose that Juliet has the exact same {\it emotional character} as Romeo. (I'm not saying J(t)=R(t), but they respond to each other in analogous ways) What would this say about parameters c and d in terms of a and b?  In this case are there values for the parameters a and b that would lead to equilibrium (i.e. all time derivatives vanish)? Is there any equilibrium solution for J(t) and R(t) besides ``complete mutual indifference", i.e. R=J=0?


%Are there values for the populations that would lead to a stable solution
%for both groups?
\end{enumerate}
{\it You might  be wondering  why we are  asking you to think about such a whimsical question like this in Phys 2210. This question is really just about developing intuitions about ODEs. Physics is full of ODEs! You will need such intuitions in almost every branch of physics. I got the question from Strogatz' ``Nonlinear Dynamics and Chaos" text.}




%
% \question  Boas  has a very useful formula for solving {\it any} linear first-order differential equation: Eq. 3.9 page 401. We will not derive this formula in class but you can check now how useful it is (8 points). For example, consider an electric  circuit containing a resistor  in series with an inductor $L$ and a source of emf $V(t)= V_0 e^{i \omega t}$. You may or may not have treated "RL" circuits in Phys 1120, but the resulting equation for the current is a pretty straightforward ODE:
%  \begin{eqnarray}
  %L\frac{d I}{d t}+ R I= V(t)
  %\end{eqnarray} 
  %Find the current $I(t)$ as a function of time, assuming $I(0)=0$.  {\it (Note: In practice, because we started with a complex voltage, you would take the real part to figure out physical currents, but no need to bother with that for now)}


\end{questions}
\end{document}

