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\begin{document}
\lhead{PHYS 2210   Fall 2011   Homework Set 3}
\chead{}
\rhead{ Due at 4 PM on \textbf{Sept 8, 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 (1 point) Boas, Chapter 8, Section 3, Problem 7.  Find the general solution.  You do not need to compare it to a computer solution.

\item Consider a softball with diameter 10 cm and mass 200 g. Use the fact that the force of air resistance on a spherical object of 
diameter D can be approximated using coefficients $c_{1}$ = (1.55$\times 10^{-4})D$ and $c_{2} = 0.22D^{2}$ (all numerical
values are in SI units). 
 
  \begin{enumerate}
  \item (1 point) Calculate the terminal speed of the softball taking into account both the linear and quadratic
terms. Show all work.
  \item (0.5 points)   The ratio of the quadratic and the linear terms of the force of air resistance can
therefore be expressed as:
 \begin{center}
 $\dfrac{|c_{2}v^{2}|}{|c_{1}v|}= \dfrac{0.22D^{2}v^{2}}{(1.55\times 10^{-4})Dv}=(1.4\times10^{3})|v|D$
 \end{center}
 Using the above ratio, 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 term?   If it were desired to approximate the effect of air
resistance on a falling softball with either the linear term or the quadratic term (not both), which
term would you keep? Explain your reasoning.
\item (1 point)  We have already seen in class an exact solution for velocity in the case where there is only linear or quadratic drag.  So, based on your answer to part b, what is the approximate speed of a softball 2  seconds after it was released from rest?
\item(1.5 points)  Let's investigate how good this approximation is.  If we keep both the linear and quadratic terms, we have a non-linear differential equation. In general we do not know how to solve this kind of differential equation, but we can solve it numerically.  Use the {\tt NDSolve} function in \emph{Mathematica} to find a the numerical solution for the velocity at  t=2 sec.   Compare this to the approximate one that you found in part c.  Was it a good approximation?
    \end{enumerate} 

\item(2 points) The Bugatti Veyron Super Sport, the world's fastest and most expensive production car, has a maximum speed of 268 miles per hour\footnote{James May noted on the BBC show ``Top Gear" that at this top speed ``the tires will only last for about 15 minutes, but it's okay because the fuel runs out in 12 minutes."}.  The engine provides maximum forward force of 12,350 N on the 1900 kg car.  Quadratic air resistance dominates, with a drag coefficient of .86 kg/m. The car starts from rest at one end of a 5 km long  test track.   Can the Bugatti reach a speed of 265 mph (very close to the maximum speed) before it runs out of road?  (Watch your units on this one!)

Problem 4 is of interest for on ongoing study on the use and effectiveness of Phet computer simulations.   If you consent to allow the researchers to have access to your homework solution to help improve the simulations, please sign the attached consent form and turn it in with your homework.   Your solutions will be anonymous to the researchers and your decision on participation in this study will not impact your grade in this class.

\item  In your textbook (section 2.4), Taylor solves for the case of a
  baseball being dropped from a high tower subject to quadratic air
  resistance, $F_D = -c v^2 \hat v$. Let's now instead look at the case of a ball
  being shot \emph{up} at an initial speed $v_0$.

\begin{enumerate}
\item(0.5 points) 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(0.5 points) Using your result from part a, 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 $v_t^2 = mg/c$.)

\item(0.5 points) Now download the PhET simulation at:
  \url{http://phet.colorado.edu/en/simulation/projectile-motion}. On
  the top right, switch the object to baseball. This sim uses
  quadratic drag: $F_D = -\frac{1}{2}c_0 A \rho_{air} v^2 \hat v$,
  where $c_0$ is the drag coefficient, $A$ is the cross-sectional area
  of the object being shot, and $\rho_{air}$ is the denstiy of air
  $=1.3 \quad kg/m^3$. (The sim shows you the value of $c_0$.) By experimenting with the sim, what initial velocity makes the ball reach the top at approximately 3 sec?
  
\item(0.5 points) Now use your formula in part b for ``time to top'' 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. How does your calculated value compare to your ``experimental" value?

\item(0.5 points) When playing with the PhET sim, does it seem to take longer for
  a ball to go from the ground to the top of a trajectory or from
  the top of the trajectory to the ground? Explain why this is the
  case.

\item(0.5 points) Again, playing with the sim, write down the initial velocity that makes the ball reach the top of its trajectory at 4 sec, then 5 sec, then 6 sec, and so on. What do you notice happening? Make a plot of $t_{top}$ vs. $v_0$ and explain in words how this relates to what you see on the sim.
\end{enumerate}

\emph{Extra Credit:} Play with the PhET sim a little more and explore anything you
  are interested in. Write down one question that you have about
  something you notice when playing with the sim.


\end{enumerate}
\end{document}  