\documentclass{article}
\usepackage{exam2}
\paper{PHYS 2210}               % <- do not include FC, FT etc

\title{Homework 2}
\time{Start of class on Thurs. Jan 20}
\semester{Spring}
\year{2011}                     % <- default is the current year
%\campus{City}
\note{ Be sure to show your work and explain what you are doing. (Correct answers, for which we cannot follow the work, are worth no credit). }
\begin{document}
\begin{questions}

 \question (4 pts) A  particle is projected with velocity $v_0$ straight up a slope which makes an angle $\alpha$ with the horizontal.  Find the time required to return to the starting point. (assume frictionless motion)  Explicitly check the units of your final formula, and discuss the special cases  $\alpha \to 0$ and $\alpha \to 90^\circ$.

 
 
 \question A soccer player kicks the ball  with a velocity $v_0$ at an  angle $45^{\circ}$ (to reach maximum range). She want to kick it in such a way that it  barely passes  on top of two   opposite team players of height $h$ (12 pts: 4+4+4).

\begin{enumerate}
 \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 Mathematica to plot $d$ as a function of $v_0$,  setting $h=2  m$. {\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.
 \end{enumerate}
 

 
\question Trajectory of a particle (8 pts: 4 each ). A particle  moves in a two-dimensional orbit defined by

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

\begin{enumerate}

\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 go the other way around?
     \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 b 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  In this problem we want to generalize  the analysis that you did in class for the  motion of a particle in   polar coordinates to spherical coordinates (4 pts each lettered part).
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 Let's investigate that the definitions given in Eqs.(\ref{coo1}-\ref{coo}) make sense.  First, 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 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 the particle is constrained 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 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  The pair of coupled ODEs

\begin{eqnarray}
\frac{dx(t)}{dt}= A x(t) - B x(t) y(t)\\
\frac{dy(t)}{dt}= -C y(t) +D  x(t) y(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)
\begin{enumerate}
\item Which of the variables, $x(t)$ or $y(t)$, represents the predator?
Which represents the prey? What reasons do you have for your choice?
\item What do the parameters $A, B, C,$ and $D$ represent?
Why do you say so?
\item 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 an ecology question 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.}



%
% \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}

