Question $3$: $(25\%)$

Please read the Newell's car$-$following model $($enclosed$).$ The author skipped some details of certain

equations. Please drive the following equations step by step:

Eq$.(4)$: $(10\%)$

Some people would prefer to describe the macroscopic behavior in terms of flows $q$ and den$-$

sities $k$ rather than velocities and spacings. A "stationary flow" will be interpreted here as some

region in the $x,t$ plane where all vehicles are traveling at nearly the same velocity $v,$ but possibly

random spacings and headways.

If

$s_n=d_n+v{}_n$

and all vehicles have the same velocity, then

$\frac{?}{\text{b}\text{a}\text{r}}(s)\frac{?}{b}=ar(d)+\frac{v}{b}ar()$

The $k$ is interpreted as $\frac{1}{\frac{?}{?}}$bar $(s)$ and the $v$ can be interpreted as $\frac{q}{k}.$ Thus

$q=\frac{1}{(\frac{?}{\text{b}\text{a}\text{r}}())}-\frac{(\frac{?}{\text{b}\text{a}\text{r}}(d))}{(\frac{?}{\text{b}\text{a}\text{r}}())}k$

Eq$.(7)$: $(15\%)$

The $x_n(t+{}_n)$ can be represented as

$x_n(t+{}_n)=x_n(t)+{}_n{v}_n(t+T_n)$

~$=x_n(t)+{}_n{v}_n(t)+{}_n{T}_n{a}_n(t)$

with $v_n(t)$ the velocity of the $n$th vehicle at time $t$ and $a_n(t)$ its acceleration. The form $(5$a$)$ follows

from the "mean value theorem" of calculus for $T_n$ of some value between $0$ and ${}_n$ but, if the

function is smooth, the value of ${}_n$ should be approximately

$T_n=\frac{{}_n}{2}$

Similarly $(5$b$)$ follows from an expansion of the $v_n$ in $(5$a$),$ but again the value of $T_n$ should be

as in $(6).$

Substitution of $(5$a$)$ in $(1)$ gives

$v_n(t+T_n)=\frac{1}{{}_n}[x_{n-1}(t)-x_n(t)]-\frac{d_n}{{}_n}$