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

\title{Homework 4}
\time{Start of class on Thurs. Feb 3}
\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 Consider a sports car which 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.  When you write Newton's law, if you are interested in finding v(x) (rather than v(t)), there is a nice trick, 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 
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 If the blocking car was removed when the car had slowed to a velocity of $200$ Km/h  and then the driver starts accelerating again with constant maximum force,  what is the velocity that the Aero TT reaches after it travels a distance of 2 Km from the point it started re-accelerating? (Briefly, discuss) 
\item Make a rough sketch (by hand, not with Mathematica!) of the car's position x(t) described by the ``story" of parts b and c (i.e., starting from when the driver entered the straight track, and ending at the final position after part c.) Comment briefly on interesting features of your graph (e.g., signs of slope, signs of concavity, interesting points...) 
\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
 $\gamma=\frac{1}{\sqrt(1- v^2/c^2)}$. Find the first two terms of the
Taylor series expansion of the energy, in the non-relativistic limit $v<<c$.  What is the second term? Does this make sense? 


\question Find the regulation size (and mass, while you're at it) for a golfball. 
\begin{enumerate}
\item The drag coefficient $c_D$  for a golfball is (very roughly)  0.3.  (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 golf ball traveling in air, and write down the 
equations of motion required to solve for x(t) and y(t) of a golfball with (just) quadratic drag. 
\item  In class we did a MMA Tutorial using NDSolve. (The code/worksheet is still available on our course calendar for Tues Jan 25)  Modify your code so that you have {\it quadratic} drag.  For simplicity and concreteness, set the angle to 45 degrees, and pick an initial speed of  80 mi/hr (converted to SI metric, of course)  Plot the trajectory for us. What is the ideal (drag-free) range in this case, and what does your code tell you the range is including quadratic drag? (Don't hunt for the optimum range like we did last week, just 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 Astronaut Alan Shepard on Apollo 14 brought a golf club with him (!) Assuming he hit it at 45 degrees (and assuming he hit it at that same 80 mi/hr, a modest swing on earth, maybe optimistic on the moon, given his clumsy space suit), calculate how far the ball landed from him, and compare to what it would have been on earth. (Which effect is more significant in the end, the loss of air drag on the moon, or the difference in gravity? )

\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}



\vspace{.2 in}


{\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}

