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\begin{document}

\title{Math 142-2, Homework 5}
\author{Your name here}
\date{\today}
\maketitle

\section*{Problem 57.2}
\textbf{57.2. Suppose a velocity field is given:
$$u(x, t) = \frac{30 x + 30 L}{15 t + L}$$}

\textbf{(a) Determine the motion of a car which starts at $x = L/2$ at $t = 0$. [Hint:
Why does $dx/dt = (30x + 30L)/(15t + L)$? Solve this differential equation. It
is separable.]}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\vspace{1cm}
\textbf{(b) Show that $u(x, t)$ is constant along straight lines in the $x-t$ plane, but the
car does not move at a constant velocity.}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\section*{Problem 57.3}
\textbf{Suppose that the velocity field $u(x, t)$ is known. What mathematical
  problem needs to be solved in order to determine the position of a car at
  later times, which starts (at $t = 0$) at $x = L$?}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\section*{Problem 57.5}
\textbf{Suppose that $u(x, t) = e^{-t}$.}

\textbf{(a) Sketch curves in $x-t$ space along which $u(x, t)$ is constant.}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\vspace{1cm}
\textbf{(b) Determine the time dependence of the position of any car.}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\vspace{1cm}
\textbf{(c) In the same $x-t$ space used in part (a), sketch various different
  car paths.}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\section*{Problem 57.6}
\textbf{Consider an infinite number of cars, each designated by a number
  $\beta$. Assume the car labeled $\beta$ starts from $x = \beta$ ($\beta > 0$)
  with zero velocity, and also assume it has a constant acceleration $\beta$.}

\textbf{(a) Determine the position and velocity of each car as a function of time.}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\vspace{1cm}
\textbf{(b) Sketch the path of a typical car.}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\vspace{1cm}
\textbf{(c) Determine the velocity field $u(x, t)$.}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\vspace{1cm}
\textbf{(d) Sketch curves along which $u(x, t)$ is a constant.}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\section*{Problem 59.2}
\textbf{Suppose that at position $x_0$ the traffic flow is known, $q(x_0, t)$, and varies with
time. Calculate the number of cars that pass $x_0$, between $t = 0$ and $t = t_0$.}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\section*{Problem 59.3}
\textbf{In an experiment the total number of cars that pass a position $x_0$ after $t = 0$,
$M(x_0, t)$, is measured as a function of time. Assume this series of points has
been smoothed to make a continuous curve.}

\textbf{(a) Briefly explain why the curve $M(x_0, t)$ is increasing as $t$
  increases.}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\vspace{1cm}
\textbf{(b) What is the traffic flow at $t = \tau$?}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\section*{Problem 60.1}
\textbf{Consider a semi-infinite highway $0 \le x < \infty$ (with no entrances
  or exits other than at $x = 0$). Show that the number of cars on the highway
  at time $t$ is
$$ N_0 + \int_0^t q(0,\tau) \,d\tau, $$
  where $N_0$ is the number of cars on the highway at $t = 0$. (You may assume
  that $\rho(x, t) \to 0$ as $x \to \infty$.)}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\section*{Problem 60.2}
\textbf{Suppose that we are interested in the change in the number of cars $N(t)$
between two observers, one fixed at $x = a$ and the other moving in some
prescribed manner, $x = b(t)$:
$$ N(t) = \int_a^{b(t)} \rho(x,t) \,dx $$}

\textbf{(a) The derivative of an integral with a variable limit is
$$ \frac{dN}{dt} = \frac{db}{dt} \rho(b,t) + \int_a^{b(t)} \px{\rho}{t}\, dx. $$
(Note that the integrand, $\rho(x, t)$, also depends on $t$.) Show this result
either by considering $\lim_{\dt \to 0} [N(t + \dt) - N(t)]/\dt$ or by using the
chain rule for derivatives.}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\vspace{1cm}
\textbf{(b) Using $\px{\rho}{t} = -\px{}{x}(\rho u)$ show that
$$ \frac{dN}{dt} = -\rho(b,t) \gp{u(b,t) - \frac{db}{dt}} + \rho(a,t) u(a,t).$$}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\vspace{1cm}
\textbf{(c) Interpret the result of part (b) if the moving observer is in a car
  moving with the traffic.}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\section*{Problem 61.3}
\textbf{Assume that a velocity field, $u(x, t)$, exists. Show that the
  acceleration of an individual car is given by
$$ \px{u}{t} + u \px{u}{x}. $$}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\section*{Problem 63.6}
\textbf{Assume that $\hat{u} = \hat{u}(\rho)$. If $\alpha$ equals a car's
  acceleration, show that
$$ \alpha = - \rho \frac{d\hat{u}}{d\rho} \px{u}{x}. $$
  Is the minus sign reasonable? Note that I have given $\hat{u}$ and $u$
  different names to make their different functional dependence explicit.  In
  particular, $u(x(t), t) = \hat{u}(\rho(x(t), t))$.}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\section*{Problem 63.7}
\textbf{Consider exercise 61.3. Suppose that drivers accelerate such that
$$ \px{u}{t} + u \px{u}{x} = -\frac{a^2}{\rho} \px{\rho}{x}, $$
where $a$ is a positive constant.}

\textbf{(a) Physically interpret this equation.}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\vspace{1cm}
\textbf{(b) If $u$ only depends on $\rho$ and the equation of conservation of
  cars is valid, show that
$$ \frac{du}{d\rho} = -\frac{a}{\rho}. $$
Note: You may assume $\frac{d\hat{u}}{d\rho} < 0$.}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\vspace{1cm}
\textbf{(c) Solve the differential equation in part (b), subject to the condition that
 $u(\rho_{\max}) = 0$. The resulting flow-density curve fits quite well to the
  Lincoln Tunnel data.}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\vspace{1cm}
\textbf{(d) Show that $a$ is the velocity which corresponds to the road's capacity.}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\vspace{1cm}
\textbf{(e) Discuss objections to this theory for small densities.}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\end{document}

