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\begin{document}
\lhead{PHYS 2210   Fall 2011   Homework Set 2}
\chead{}
\rhead{ Due at 4 PM on \textbf{Sept 1, 2011}}

%\maketitle

Show and explain all of your work!   Correct answers for which we cannot follow your work are worth no credit.


\begin{enumerate}

\item Point P sits on the rim of wheel of radius $\rho$ that is rolling on the ground in the positive x direction. Its position vector can be described by the sum of a vector $v_{1}$ for circular motion about the center of the wheel plus a vector $v_{2}$ that describes the horizontal motion of the center of the wheel along the line $y=\rho$ at constant velocity.  In other words, its position at time t is given by
\[  \vec{r}=   \vec{v_{1}}+\vec{v_{2}} = (\rho \sin{(\omega t)} \hat{x} +   \rho\cos{(\omega t)} \hat{y})+ (\rho \omega t \hat{x} +\rho \hat{y}) \]

\begin{enumerate}
\item[a)] (1 point) Find an expression for the velocity of Point P.  
\item[b)] (1 point) Are there any times when the velocity is 0?  Where is Point P on the wheel at these times (on the top, on the front edge, etc)?  Does this make sense?
\end{enumerate}

\item  A slingshot launches an angry bird\footnote{Apparently the bird is so upset over some stolen eggs that it can't fly.} up in the air with a speed $v_{o}$ at an angle of $B$ (with respect to the ramp) up a hill of with an incline of $H$ above the horizontal.

\begin{figure}[h] 
   \centering
   \includegraphics[width=3in]{hw2fig.jpg} 
\end{figure}

\begin{enumerate}

 \item[a)] (1 point) Show that distance \emph{along the hill} where the bird lands can be expressed as  $\dfrac{2 v_{0}^{2} \sin{(B)} \cos{(B+H)}}{g \cos^{2}{(H)}}$.
 (Hint: use tilted coordinates where one axis is along the slope of the hill, and some angle addition identities might be helpful here.)
\item[b)](1 point) For a given $H$, at what angle should the bird be launched to achieve the maximum distance?  
\item[c)](1 point) Show that the expression for the maximum distance is $d_{max}=\frac{v_{o}^{2}}{g(1+sin(H))}$
\item[d)](1 point) For $H = 20^{\circ}$, use \emph{Mathematica} to make a plot of the expression you obtained in part a from $B$=0 to 90$^\circ$.  Attach the plot to this assignment, and verify that the maximum distance value occurs at the angle given by your answer to part b. 
\end{enumerate}

\item (2 points)  Taylor Problem 1.48.  Note that the problem is asking you to express $\hat{\rho}$, $\hat{\phi}$, and $\hat{z}$ in terms of their $\hat{x}$, $\hat{y}$, and $\hat{z}$ components, but these expressions may include the coordinates $\rho$, $\phi$ and z.


\item A honeybee exits its hive and flies in spiral path. Its position \textbf{r} at time $t$ is given in polar coordinates by
\[   r = be^{kt}  \hspace{1in}  \phi = ct   \]
where $b$, $c$, and $k$ are constants.
\begin{enumerate}
\item[a)](1 point) Use the expressions that we found for \textbf{\.r} and \textbf{\"{r}} in polar coordinates to find the velocity and acceleration vectors for the bee.  Are the velocity and acceleration vectors orthogonal to each other?
\item[b)](1 point) Show that the angle between the velocity and acceleration vector remains constant as the bee moves away.  
\end{enumerate}





\end{enumerate}
\end{document}  