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

\title{Homework 5}
\time{Start of class on Thurs. Feb 16. }
\semester{Spring}
\year{2012}                     % <- default is the current year
%\campus{City}
\note{ Feb 16 is also the day of our first first exam!  You may bring one side of one page of paper with your own {\it hand written} notes, if you like. Please read the LAST homework question first, so you know it's coming. This may require some planning on your part. }
\begin{document}

%\textcolor{red}{Overall this homework appears short, because the first test was given a week earlier (the day after this homework was due). Will we cover rockets? See question \#4; added by DC.}

\begin{questions}


\question  {\it Taylor series are the single most important and common approximation technique used throughout physics. We need to keep practicing with them! }
\begin{enumerate}
\item  Suppose $f(x)=1/(1+x)$. Find an approximate expression for $f(x)$, going out to second order. Use it to estimate f(0.1) and f(5). In both cases, compare the exact result for $f(x)$ with your series approximation, going to first order (terms proportional to x) and also second order ($x^2$). Does your estimate improve at second order in both cases? Comment on the general criterion about x you would guess tells you when the series approach seems to be a fruitful one here. 
\item Now let $f(x)=(1-x)/(1+x)$. Taylor expand to approximate $f(x)$ out to second order. There are many ways to do this. First, use the usual formal formula for Taylor expansion. Another way is to take your series answer in part a, and multiply it by (1-x). Try it this way too, and use it to check yourself, both methods should agree to all orders! Which method do you prefer?
\item  Now let $f(x)=(a-x)/(a+x)$, where a is a constant with units, like $a=5 m$. Expand to second order. In this case, do NOT start ``from scratch" blindly using the math formula. Instead, factor out ``a", so that it looks {\it exactly} like what you had in part b, except the ``thing you're expanding in" isn't x. What is it?   For this part, what is now the general criterion on x that tells you when this series approach seems to be a fruitful one?  Why? (And please do not say ``x must be small" or ``$x<<1$". Small compared to what? You cannot compare meters to numbers, or apples to oranges!) 
%
{\it (Once you understand this trick you will be spared having to ever painfully Taylor expand much of anything - the front flyleaf of your text will tell you how to expand almost anything you'll ever encounter)}
\item Taylor's eqn 2.33 gives the velocity of a ball that has been dropped from rest with linear drag.  Obtain an approximate expression for $v_y$(t) in the case of small drag.  Include terms to 2nd order (Feel free to use the front flyleaf to save some grief!)   What is the physical meaning of the leading nonzero term in your approximate solution? What is your interpretation of the {\it sign} of the next term?  As in the previous question, what exactly do we mean by ``small drag" here - what exactly is small, compared to what, here?  

 \end{enumerate}


\question
\begin{figure}[htc]
\centering
{\includegraphics[width=2.0in]{Hom5fig1.eps}}
\end{figure}

Danny Jr. (mass $m_d$) is  standing at one edge of a 4 m long canoe (mass $M_c$) (See Fig. 1). He observes a turtle on top of a rock standing close to the other edge of the canoe. Danny wants to catch the turtle and starts walking towards it. Ignoring water friction:
\begin{enumerate}
\item Qualitatively describe  the motion of the system (canoe+ Danny) as he walks forward.  If initially the canoe is 3m away from the dock, where  is Danny with respect to the dock when he reaches the other end of the canoe? (We want a formula in terms of $m_d$, $M_c$, and the given starting dimensions) 
%\item If $m_d=30$ Kg and $M_c=30$ Kg and Steve can stretch his arm 1m away the  canoe edge, can he catch the turtle?
\item If $m_d=30$ kg and Danny can stretch his arm 1 m away from the canoe's edge, what is the minimum mass the canoe must have to ensure that he can catch the turtle? (Also, what happens in the limit of a very light canoe? A very massive canoe?) 
\end{enumerate}


%\question A flat plate of mass M  has a triangular shape with the dimensions shown in Fig.2. The plate has a uniform density ( mass per unit area in this case).  
%\begin{figure}[htc]
%\centering
%{\includegraphics[width=2.5in]{Hom5fig2.eps}}
%\end{figure}
%\begin{enumerate}
%\item Based on your physical and mathematical intuitions, without calculating, where do you predict the center of mass is, and why?
%\item  Now mathematically determine its center of mass coordinates. Does your answer match your intuitions from part a? Briefly, explain (or reconcile!)
%\end{enumerate}


          
 \question  Exam review!  (The rest of this homework is required, and you should turn it in. But, for obvious reasons, the grader will not look over what you write in {\it detail}, each part will be graded simply ``credit/no credit")  Our exams will likely be 4 or 5 questions (each multiple parts), so we figure $\sim$15-20 min/question.  For the question below: if you have a way to scan or input your work -  look for the discussion forum for this problem on our D2L site and upload it! (This isn't required, but would be helpful. )




On our website, in the syllabus, after our ``Introduction" paragraph, there is a link to our course goals. Go there, read that page. Then -
\begin{enumerate}
\item  Find in the textbook (Taylor, or one of the ``recommended" texts available in the Physics library reserve),   or invent  a plausible exam question covering material we have covered so far this term  (You might also explicitly think about what we've covered in Boas, too - can you ``hook that in"? ) Write it  down.
\item   Solve it. (Include your solution with your homework) (If it takes $<\approx$ 5 min, how could we have made it a little more interesting/richer/challenging? If it takes $>\approx$ 20 min, how could we scaffold/hint/simplify to get at the interesting physics, without making it tedious/grungy?)

\item Write a brief (just a few words) summary which characterizes what {\it content} this problem covers (e.g, ``position vectors in polar-coordinates", or ``Newton's law in tilted Cartesian coordinates", or etc)

\item Which of  the course scale learning goals does your problem address (list them by number)  
See \url{http://www.colorado.edu/physics/phys2210/phys2210_sp12/course_goals.html}
\end{enumerate}


% \item Repeat the above, for Chapter 1 (or Chapter 3, at least what we've covered so far)  material.

%\textcolor{red}{Below is Alysia's HW5 \#5 (DC opinion: needs more prediction and sense-making)}

%\question

%Consider a rocket with initial mass $m_0$ that travels vertically. The rocket ejects spent fuel at a constant velocity ($v_{ex}$) relative to the rocket's motion.

%\begin{enumerate}[(a)]

%\item If this rocket is designed such that the rate of fuel ejection is constant (i.e., $\dot{m} = - k$), show that the equation of motion for this rocket is,
%\begin{equation}
%m \dot{v} = k v_{ex} - mg
%\end{equation}
%Solve this differential equation for $v(t)$ using separation of variables. You will assume that $g$ doesn't change appreciably.
%\item Describe what happens to the rocket if the value of $k v_{ex}$ was smaller than the initial value of $m g$
%\item Show that for a rocket that starts from rests at $y = 0$ the resulting expression for $y(t)$ is,
%\begin{equation}
%y(t) = v_{ex} t - \frac{1}{2}g t^2 - \frac{m v_{ex}}{k} \ln \left(\frac{m_0}{m}\right)
%\end{equation}
%(Hint: The following integral might be useful: $\int \ln(x) = x\ln(x)+x+const$.  
%f you are interested, you can integrate $\ln(x)$ by parts to obtain this result.)
%\item  Assume that the rocket burns fuel for 200 seconds. Use your favorite computational tool to plot the velocity and position of the rocket as a function of time (up to 200 sec). Let the initial mass of the rocket be 2$\times$10$^6$ kg, the rate of mass ejection be 8333.33 kg/s and the exhaust speed be 3000 m/s. Obviously, the rocket starts from rest. How high is the rocket after 200 seconds? What does this tell you about your assumption about g in part (a).
%\item Real rockets experience to air drag. Due to their size and typical speeds, the air resistance is best modeled as quadratic (turbulent). Consider your plots in part (d). Describe how your plots might change by adding $bv^2$ drag to the model.
%\item Using your favorite computational tool numerically integrate the equation of motion which includes air drag. Use the same parameter values as in part (d). Assume the drag coefficient ($b$) for the rocket is 0.80. Plot both the velocity and vertical position of the rocket  as functions of time. Compare these plots to your predictions in part (e) and your analytic results in part (d).
%\end{enumerate}


\question  Get together with at least one other person from class, and share your made-up exam question. Do theirs, and discuss with them whether you think they got the level and coverage right.  (If you can't find someone in person, use our D2L forum which I created for this purpose - but don't wait till the last minute, get this arranged with plenty of time) 
To turn in to us:
\begin{enumerate}
\item Write down the name(s) of the person you worked with
\item To show us that you really did this activity (yes, this IS a homework problem, for credit!), write down in your homework the problem of theirs that you did. (If it's really long, paraphrase)
\end{enumerate}


%{ \it Extra credit question for this week: }
%A collision  between two bodies is defined to be elastic if the total kinetic energy before and after the collision is the same. Consider an elastic collision between two identical atoms. One moves with velocity $\vec{v}_1\neq 0$. the other starts at rest $\vec{v}_2= 0$. Denoting  $\vec{v}'_{1,2}$ the corresponding velocities after the collision:
% Write down the vector equations representing  the conservation of momentum and the scalar equation representing the conservation of kinetic energy in an elastic collision. Use this to prove that the angle between  $\vec{v}'_{1}$ and  $\vec{v}'_{2}$ is $\pi/2$. Can you think of a situation where this might prove to be useful?

    %      {\it  We have not yet begun  our discussion  of energy conservation. Nevertheless  the  knowledge you have from prior years should be  enough to solve this problem.}
\end{questions}
\end{document}

