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

\title{Homework 3}
\time{Start of class on Thurs. Feb 2}
\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 part }
\begin{document}
\begin{questions}


\question	Consider a ball that moves vertically under the influences of both gravity and air resistance.  For the purposes of this problem, take vertically upward as the positive direction.  For each equation of motion below, determine whether that equation applies to (i) a situation in which the ball moves upward, (ii) a situation in which the ball moves downward, (iii) either of these, or (iv) neither of these.  Explain your reasoning for each case. (4 pts total on this one)

a) $	m (dv_y/dt) = - mg - c_1 v_y,$\hspace{2 cm} b)   $m (dv_y/dt) = - mg + c_1 v_y$

c) $	m (dv_y/dt) = - mg - c_2v_y^2,$\hspace{2 cm} d) $m (dv_y/dt) = - mg + c_2v_y^2$



\question  You're hanging out by a campfire after acing the first Phys 2210 exam. You notice a tiny piece of fly ash that is ejected from the fire vertically upward with initial speed $v_0$. %We might want to predict if the ash will reach tree-level! 
Since physicists tend to love to think about casual observations like this more formally,  you decide to try to predict if the ash will reach tree-level. 
 Let's measure the position of the fly ash, $y$ from the point of release, once again taking vertically upward as the positive direction. 

\begin{enumerate}
\item Clearly the physics of ``flying ash" could be very complicated!  But, let's start with the {\it simplest possible} model and go from there. If you consider ash to be a point particle of mass m acting only under gravity, what is the equation of motion? What is the maximum height (in terms of $v_0$ and g)? 
\item The next simplest model might be to also include simple air drag. Taylor discusses the choice between ``linear" and ``quadratic" drag. Knowing that ash particles are extremely tiny, and not particularly fast moving to start with, if you had to choose just ONE of these, which should it be? Briefly (qualitatively is all we want) justify your answer. 
\item Now, using this model, find the time for the fly ash to reach its highest point and its position $y_{max}$ at that time. {\it (Note that velocity $v_y(t)$ and position $y(t)$ are derived in the lecture notes, and the text. Of course, it's always a good idea to rederive them yourself, but not required for credit. However, you should be careful, because in the text, Taylor defines +y to be the other way, so watch your minus signs!  I suggest using notation from question 1 to define symbols.  No numbers here, do it all symbolically.) }

\item Show that as the drag coefficient approaches zero your answer in part c reduces to the well known freshman physics result you got in part a.  {\bf Hint:} If the drag coefficient is  small  the terminal velocity is big,  so $v_0/v_{ter}$ is very small. Use  the Taylor expansion $\ln(1+\epsilon)\approx \epsilon-\epsilon^2/2+\dots$ for $\epsilon\ll1 $.
\end{enumerate}

%\question  How long does it take for a cyclist to slow down due to air resistance. To estimate that you might first go %to wikipedia and find out that the coefficient of quadratic air resistance on a cyclist is around $c_2=0.2$ %N/(m/s)${}^2$(12 pts: 4 each).
%\begin{enumerate}
%\item Does this value of $c_2$ make sense? To answer  that assume the total mass (cyclist  plus cycle) is  $m=80$ Kg and compute the characteristic time $\tau$ assuming an initial speed $v_0\sim 20$ m/s. Discuss.
%    \item How long will it take to the cyclist to slow down  from $v_0$ to $v= 15$m/s and $v=5$m/s.
%    \item Can you use this reasoning to determine  how long it will take the cyclist to stop?
%\end{enumerate}

\question A grain of pollen (mass m) slides down a smooth, flat, tilted solar panel (tilted at $\theta$ from horizontal) under the influence of gravity. 
\begin{enumerate} 
\item If the motion is resisted by a linear drag force $f=kmv$  find an expression for how far down the plane the grain slides in time t, assuming it is released from rest.  {\bf Hint:} Draw a free body diagram to start! 

\item Show that as the drag coefficient becomes very small, your answer in part a reduces to a simple freshman physics result (which is what?)   Comment briefly on the similarities and differences between this problem and question 2. 
{\bf Hint:} Here, I claim the useful Taylor expansion you'll want will be $e^x \approx 1+x +x^2/2 + \ldots$, for $x \ll 1$ And yes, you'll really need to go all the way out to that $x^2/2$ term!   \\

{\it For you to think about. You do not have to write this up for credit, but we'll revisit this idea again this term, it's very important!}  The power x in $e^x$ must be unitless.  ``Drag coefficient'' has units. So it can not be exactly correct to say the ``drag coefficient becomes small", without answering ``small compared to what"? What exactly is it that is really the ``small thing, x'' in this case? 
\end{enumerate}

% \question  Boas  has a very useful formula for solving {\it any} linear first-order differential equation: Eq. 3.9 page 401. We casually sketched the derivation of this formula in class, but you can check now how useful it is. For example, later this term we'll be studying ``driven harmonic oscillators", but there's a simpler case where a free (but ``damped") particle is being wiggled by an external force, f. The resulting equation for velocity is a pretty straightforward ODE: 
 %\begin{eqnarray}
 % m\frac{d v}{dt} - bv = f_{ext}(t) = f_0 e^{i \omega t}
 % \end{eqnarray}
% Look at the equation and physically interpret,  in your own words, each of the two terms on the left side of the exression.  Then, use Boas to find velocity $v(t)$ as a function of time, assuming $v(0)=0$.
 
% {\it Note: The right side is complex! That may strike you as rather odd right now if you've never seen that before. We'll talk much more about it soon. We'll interpret this using a wonderful theorem, ``Euler's theorem", which relates complex exponentials to simple oscillatory sin's and cos's.   In practice you have to take the real part of the answer to figure out physical velocities, but no need to bother with that for now!}


  



\question In Season 1 Episode 4 of Mythbusters (originally aired in 2003), Jamie and Adam considered the myth that a penny dropped from the Empire State building can kill a person walking on the sidewalk below. You can watch a clip from this episode here: \url{http://youtu.be/PHxvMLoKRWg}.  Let's do our own investigation! 
%%\\In this problem, you will answer the question: ``Did they really {\it have} to do the experiment?''

\begin{enumerate}
\item Look up the size and mass of a penny.  Using the drag coefficients given by Taylor pp 44-45, which means we are assuming a spherical penny $\ddot\smile$, find the terminal speed of a dropped penny, taking into account both linear and quadratic terms together. Then, write down the differential equation for the motion of a falling penny (Keep both linear and quadratic terms. Clearly articulate your sign conventions) 

\item For what range of speeds will (i) the linear term of air resistance dominate
 over the quadratic term? (ii) the quadratic term dominate over the linear?  Discuss the relative importance of linear and quadratic drag here  - if you had to pick just ONE term (linear or quadratic), which would you use, and why? 

\item If we keep both linear and quadratic terms, we have a non-linear differential equation which we can only solve numerically.  Use your favorite numerical differential equation solver to determine and plot velocity and position of the penny as a function of time, as it falls from the top of the Empire State building (381 m).  Also find the time it hits the ground, and the speed with which it hits.  Include your code and  plots with your homework.\\
Mathematica users might find this screencast on using {\tt NDSolve} (Mathematica's built-in numerical differential equations solver) helpful:
\url{http://youtu.be/zKO6v0w0KdI}\\
{\it {\bf Be Elegant:} You could use Mathematica's {\tt EventLocator} method to stop the integration when the penny hits the ground. Helpful link for that:} \url{http://goo.gl/Mkz3i}

\item Compare the  result for ``final velocity" from your numerical results (part c) with what you got  in part a, and also what you get by assuming JUST the one dominant drag term you chose in part b. Comment. 


 \item  Now for the interesting part - could this penny kill someone? Use basic freshman physics and make crude (but quantitative) estimates to discuss whether you think the myth is busted or not!  {\it (I can think of many physics approaches to evaluate what might injure you  - think about it a bit! Just e.g., you might assume a head applies a constant force to stop the penny during impact. If so, we've got about 1mm of tissue to stop a penny before it drives into the skull. Or, how about this -  ``Wikipedia: hydrostatic shock" says a skull pressure of  $10^7 N/m^2$ can seriously injure you. Could that be useful?) } 
 
 
\end{enumerate}
%%% DC Commented out Danny Rehn's problem, suggest replace with above NDSolve problem, comments? %%%%

%\question In your textbook (section 2.4), Taylor solves for the case of a baseball being dropped from a high tower using quadratic air resistance, $F_D=-c v^2\hat v$. LetÕs look at the case of a ball being shot {\it up} at an initial speed $v_0$.
%\begin{enumerate}
%\item Draw a free body diagram for a ball moving vertically upwards, subject to quadratic air drag. Write down a differential equation for this situation, and solve this differential equation for v(t). Make a rough sketch of v(t) vs. t, and briefly discuss any key features.
%\item Using your result from the previous part, find an expression for the time it takes to reach the top of the trajectory. (It will look simpler if you write it in terms of terminal velocity, which  satisfies $v_t^2=m g/c$)
%\item Now download the PhET simulation http://phet.colorado.edu/en/simulation/projectile-motion. 
%
%On the top right, switch the object to Ôbaseball.Õ This sim uses quadratic drag: $F_D = -{1\over2} c_0 A \rho_{air}v^2\hat{v}$, where $c_0$ is called the drag-coefficient, A is the cross-sectional area of the object being shot, and $\rho_{air}$ is the density of air = 1.3 kg /m$^3$. (The sim shows you the value of $c_0$ and diameter it has picked for a baseball, on the right side of the simulation) Use your formula in part b for "time to top" with these numbers to deduce what numerical initial velocity $v_0$ you need to get the ball to reach the top of its trajectory at precisely t=3 sec.  Now test it, you can input $v_0$ into the sim and fire the cannon.  Aim the cannon at 90$^o$ (or 88-89$^o$ if it is easier to see the trajectory) and switch on air resistance. The little + and - glasses let you zoom in or out.  Does the ball reach the top at t=3 sec? (It should!)
%\item When you fired the ball on the PhET sim, did it take longer for the ball to go from the ground to the top of the trajectory or from the top of the trajectory to the ground? Explain why this is the case.
%\item Now let's look at another interesting feature of shooting an object up in the air. Start increasing the value of $v_0$ in the sim. Double it from what you had before, then increase it by 10, and then by 100. What is happening to the time to reach the top? Use your formal mathematical results from above to explain what is happening!
%
%\item  Play with the PhET sim a little more and explore anything you are interested in. Write down one question that you have about what you notice while playing with the sim. 
%
% \end{enumerate}

 \question Consider a ball  thrown at an angle $\theta$  above the horizontal ground with an initial speed $v_0$ in a medium with linear drag.  For numerical solutions, computational tools don't deal well with unknown symbols (!), so let's consider a particular case where $v_0=v_{ter}$ and $v_{ter}^2/g=1$. We know that in vacuum the maximum range is at $\theta =\pi/4$.
Let's try to estimate the maximum range (and angle) when we include air resistance:

\begin{enumerate}
\item Plot Eq.(2.37) in Taylor for different values of $\theta$ and try by inspection to figure out the angle at which the range is maximum.  (With the assumptions above, most unit-full quantities become ``one'', but do be careful of the fact that e.g. $v_{0,x}$ is $v_0\cos(\theta)$, not $v_0$), etc. Briefly comment on real-world implications!
\item Use a root finder to find the range when $\theta=\pi/4$. Any root finder will require you to make a guess for the root. This is what is called the ``neighborhood of the root'' or ``bracketing the root''. A screencast showing how to use Mathematica to find roots is available here: \url{http://youtu.be/673IQ6Z-6Yc}
\item Repeat for different values of $\theta$ (homing in on a small range near the angle you estimated in part a). Continue until you know the maximum range and the corresponding angle  to two significant figures. Connect to point a) and compare with the ideal vacuum values.  Briefly, discuss.
    
    
     {\bf Extra credit}  Check what happens in the limit $v_0\gg v_{ter}$ (say $v_0=100 v_{ter}$) and $v_0\ll v_{ter}$ (say $v_0=0.01 v_{ter}$). Discuss if your results make sense. (+4 points)
\end{enumerate}
 
{\bf Note:} Judiciously use ``print selection" on a minimalist subset  of your MMA notebook so we can see your results, and discussion, without having to wade through pages of preliminary plots and calculations. \end{questions}
\end{document}

