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

\title{Homework 4}
\time{Start of class on Thurs. Feb 9}
\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 A car is braking hard. There are two significant resistive forces acting on it, a quadratic ($c v^2$) air drag, and a constant ($\mu mg$) frictional force.  If you are interested in finding v(x) (rather than v(t)), there is a commonly used method known as the ``v dv/dx rule", which uses the chain rule to rewrite $\dot{v} = {dv\over{dt}} = {dv\over{dx}} {dx\over{dt}} = v {dv\over{dx}}$.
\begin{enumerate}
\item  Write down the equation of motion for $\dot{v}=f(v)$ and use the ``v dv/dx" rule to solve the equation of motion directly for v(x), and show that the distance the car needs for a full stop is:
    
   \begin{equation}
    x_{max}=\frac{A^2}{2 \mu g} \ln\left(\frac{A^2+v_0^2}{A^2}\right)
    \end{equation} here $\mu$ is the friction  coefficient. What is the constant $A$ in this case ( in terms of given parameters in the differential equation?) 


\item 
My car has mass $m \approx $ 1600 kg, and (I estimate) $c \approx$ 0.6 kg/m. I'm driving down I-70 towards Moab, traveling 75 mph, when I see a stopped car on the highway 300 m front of me. I slam on the brakes! Assuming a friction coefficient of $\mu=0.7$, find  my stopping distance (do I crash, or am I safe?), and then compare it to what it would have been in the absence of any air  drag (i.e. just the road friction alone). Briefly, comment. If I'd been driving a sports car, traveling at twice this speed (but with all other parameters the same) would air drag have made a more significant difference? Again, comment! 
%The SSC Ultimate Aero TT,  one of the world's fastest production
%cars (at a cool \$600,000+), has a maximum speed of 412 Km per hour. The engine provides
%maximum forward force of 11260 N on the 1200 kg car.  For this car, $c$ =0.86 kg/m. On a race  track, the car
%exits a turn with a speed of $v_0=300$ km/hr.  As soon as the driver  enters the straight track after the turn,  she realizes there is another car blocking  the track  2 km away. The driver slams on  the brakes. Assuming a  friction coefficient $\mu =0.7$, 
%use your above result to compute the distance required for the Aero TT to stop. Compare this distance to  the corresponding distance in the absence of air drag (i.e. just kinetic road friction).  (Briefly, discuss)

\item Car engines are complicated, they are not simple ``constant force" devices, but I estimate that at highway speeds my car might apply a fairly steady forward force against the road of about 1500 N.  If the stopped car managed to pull off to the side of the highway when I had slowed to $30$ mph  and I immediately started re-accelerating with this constant maximum forward force,  how much highway distance would I need to get my car back up to 75 mph? (Include the quadratic air resistance, of course!)  {\it Hint: the v dv/dx rule may be useful again}

%\item What is the terminal velocity of a Prius according to these estimates? (Does that seem reasonable? Are we neglecting anything important in our estimates?)  

\item Make a rough sketch (by hand, not with Mathematica!) of my car's velocity $v(t$) described by the ``story" of parts b and c.  {\it Note: that's v(t), not v(x). Don't calculate it, just sketch what it should look like}. Comment briefly on interesting features of your graph (e.g., signs of slope, signs of concavity, interesting points...) 
\end{enumerate}

%\question  {\it The joy of Taylor Expansions. 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 $f(x)=1/(1-x)$. Taylor expand to 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 $a=5 m$. Taylor expand to second order. In this case, do NOT start ``from scratch" blindly using the Taylor 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 One astronaut is in a circular orbit around the Earth  a distance $R$ from the center of the Earth. Another astronaut is in orbit a bit closer to the Earth, at a distance $R - d$. The difference in the magnitude of the
%%gravitational field between their locations is
%%$
%%\Delta g=\frac{G M_E}{(R-d)^2}-\frac{G M_E}{R^2}
%%$
%%Use an appropriate series expansion to obtain an approximate expression for $\Delta g$. What relationship between $d$ and $R$ are you assuming? Your answer must contain one non-zero term. Does the sign of your answer seem sensible? Lastly - use your result to find the difference in g between the bottom and top of Gamow tower. Is your answer consistent with the common claim that $g=9.8 m/s^2$ anywhere near earth? Comment.
%
% \end{enumerate}
%
%% \question  
%% \begin{enumerate}
%% \item In Taylor's book, he simplifies Eqn 2.42 for the range of a projectile by assuming
%%that $R^2\sim R^2_{vac}$. Show that if instead you use the quadratic equation to solve for $R$ directly  in
%%Eqn 2.42, and make the appropriate Taylor series expansion, that you obtain
%%the expression given in Equation 2.44. 
%%
%%\item If you use your result from the previous part to compute $R$ when $v_{y0}=v_{ter}$, what do you get? Does this make sense? (Briefly, discuss) (2 pts) 
%%\end{enumerate}


  \question  Recall from special relativity that for a particle moving at a relativistic speed,
$v$, the energy $E=\gamma mc^2$, where
 $\gamma=\dfrac{c}{\sqrt{c^2- v^2}}$. 

\begin{enumerate} 
 \item In the non-relativistic limit, find the first two nonzero terms in the series expansion of the energy by using the Taylor expansion of $\dfrac{1}{\sqrt{1-x}}$. What is the non-relativistic limit? What is $x$ in this case? About what point are you expanding? What is the second term? When Einstein did what you just did (100 years ago), I think he must have smiled a big smile at that second term - why? 
 \item How can an expansion of $\dfrac{1}{\sqrt{x}}$ be made to agree with your answers to part a? Comment.\\  {\it Hint: you won't be expanding around $x$ = 0 this time! About what point should you expand?}
 \item I claim that an expansion of $\dfrac{1}{\sqrt{x}}$ about $x$ = 0 makes little sense. Why? Interpret this limit physically.
\end{enumerate}

%{\bf DC: I have a note to attempt to include real data in a rewritten version of problem 4 below.}

\question Find the regulation size (and mass, while you're at it) for a soccer ball. 
\begin{enumerate}
\item The drag coefficient $c_D$  for a soccer ball is about 0.25.  (Note: the c in $c v^2$ is given by  ${1\over2} c_D \rho D^2$) Calculate the numerical value of the quadratic drag constant ``c"  for a soccer ball traveling in air at STP, and write down the 
equations of motion required to solve for x(t) and y(t)  with (just) quadratic drag. 
\item On our course calendar, right next to this homework, you can find a sample Mathematica notebook that solves for trajectory with {\it linear air drag. }Modify that code so that you have {\it quadratic} drag.  For simplicity and concreteness, set the kick angle to 45 degrees, and pick an initial speed of  90 mi/hr (converted to SI metric, of course)  Plot the trajectory. What is the ideal (drag-free) range in this case, and what does your code tell you the range is with quadratic drag? (Don't hunt for the optimum range like last week, stick with a 45 degree angle)  Also, calculate what fraction of the initial {\it speed} is lost from when you first hit the ball (at 45 degrees) to when it lands.  (Where does the lost energy go?) 
%\item When I first started thinking about this problem, I wondered whether the fractional reduction in speed was a constant, or whether it depended on the initial velocity. What do you think? (No formal calculation required, just make a qualitative physics argument.) {\it  Check} yourself with your code, and comment. 
\item A few years ago, there was big controversy when the Soccer Federation banned international soccer competitions in cities above 2500 m elevation
% (\url{http://en.wikipedia.org/wiki/High_altitude_football_controversy}). 
(This was out of concern about players' health, and has since been eliminated.) But, what about the impact on the physics of the game? Use your code to estimate how much farther that same kick would take a soccer ball in La Paz, Bolivia (elevation, 3000 m) compared to a game at sea level (which your code was assuming.) Do you think this difference might be measurable/noticeable?  {\it Note: The air density at 3000 m is about 70\% what it is at sea level.}

\end{enumerate}




%\question  Two FBI agents (let's call them Mulder and Scully) are investigating the wreckage
%of the spaceship is in three large pieces around a northern Colorado town.
%One piece (mass = 300 kg) of the spaceship landed 6.0 km due north of the
%center of town. Another piece (mass = 1000 kg) landed 1.6 km to the southeast
%(36 degrees south of east) of the center of town. The last piece (mass = 400
%kg) landed 4.0 km to the southwest (65 degrees south of west) of the center
%of town. There are no more pieces of the spaceship. The Air Force, which
%was watching the spaceship on its radar, claims it was moving with a constant
%speed of 5 m/s to the east at a height of 1.96 km. It was 100 m west of the
%center of town when the spaceship spontaneously exploded and the pieces fell
%to the ground. There is also evidence  that none of the pieces aquired appreciable vertical velocities immediately after the explosion.  Agents Mulder and Scully think a missile hit it. Are the fragments
%consistent with the spaceship exploding spontaneously? If not, can you
%tell what direction the missile came from?

%\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 Consider a familiar horizontal spring-mass system. Recall that the the solutions for both the velocity and position of the mass are oscillatory. % (i.e., the depend on $\cos$ or $\sin$ or their sum). 

\begin{enumerate}[(a)]
\item Write down the second-order differential equation which describes the position of the mass. Then,  as we did in the numerical integration tutorial in class, write this differential equation as two first order differential equations (one for dx/dt, one for dv/dt) {\it (That in-class Tutorial is also  available for review/download on this week's course calendar) }
\item Assume the oscillator starts from rest at $x$ = +1 m, with  $m=0.1$ kg and   $k=10$ N/m.  Recalling that the period of oscillations is given by $T = 2 \pi \sqrt{m/k}$, let's pick the time step for our numerical integration to be 10\% of T (so 1 period has 10 time steps). Using the computational tool of your choice, you will be writing a {\tt for} loop to numerically integrate these equations for several periods. (For help writing {\tt for} loops in Mathematica, review the numerical integration tutorial, use the documentation or check out this screencast: \url{http://youtu.be/UpkcScYeQTc}.)\\ {\bf IMPORTANT:}\\ 
 We'd like you to write TWO codes: (code 1, ``Euler-Cromer'' method) Compute the force at an instant, update the velocity (using that force), then update the position (using that {\it updated} velocity!) and then repeat.  Alternatively, (code 2, ``Simple Euler method'') Compute the force at an instant, update the position first, then update the velocity (and repeat.)   See the difference? (Compare these two methods by using each of them to integrate the spring mass system and plot the position of the mass as a function of time.) What do you notice? \\{\it One anecdote claims the Euler-Cromer method was discovered by a high school physics student. The story is given in this \href{http://ajp.aapt.org/resource/1/ajpias/v49/i5/p455_s1?&login}{paper}.}
\item Try decreasing your integration time step by an order of magnitude. What changes do you have to make to your code? What happens to your solutions? When numerically integrating, what would help you pick the time step? (Why not, say, choose a billionth of a period in this case, to get super accuracy?) 
\item Several good integration algorithms are built in to a lot computational tools (e.g., Runge-Kutta, Dorman-Prince, etc.). Use one of these built-in integration methods to integrate the equations of motion and compare the results to those in part (b) or (c). Mathematica users can use NDSolve, a function which attempts to select the best method given the equation of motion. For help using NDSolve, check out this screencast: \url{http://youtu.be/zKO6v0w0KdI}. Notice, you don't have to pick a time step!
\end{enumerate}

%{\bf Extra credit: More fun with Taylor series} An (ideal, unstretchable, massless) string is wrapped tightly around the earth. Then, 1 extra meter is added to its length. You pull up at *one point* of the string. Approximately how high above the ground can you pull this point? (You may be quite surprised by the answer, I was!) 


\end{questions}
\end{document}

