Parameter values

Two vehicles, numbered i and $i=1,$ are travelling on a single$-$lane freeway. At time $t=5s,$

vehicle $i-1$ has a speed of $15\frac{m}{s},$ vehicle i has a speed of $y\frac{m}{s},$ and vehicle $i$ is behind

vehicle $i-1$ with a clearance of $z$ meters. Vehicle $i$ is $4$ meters long, and vehicle $i-1$ is $4\mathrm{.}5$

meters long. Assume:

$(1)$ For vehicle $i-1$ : from time $t=5s$ to time $t=8s,$ it brakes with a constant deceleration

of $-2\frac{m}{s^2}.$ After $t=8s,$ it keeps a constant speed of $9\frac{m}{s}.$

For vehicle $i$ : its movement is consistent with the GM$5$ car$-$following model, with $m=1,$

$n=0\mathrm{.}6,$ and ${}_5=5sec*$ meter ${?}^{-1\mathrm{.}5}$

$a_i(t)={}_5\frac{[v_i(t){]}^m}{[x_{i-1}(t)-x_i(t){]}^n}[v_{i-1}((\frac{?}{\text{b}\text{a}\text{r}}(t)))-v_i(t)]$

The values of $y$ and $z$ are provided in the attached file "Assignment $2-$ parameters.pdf$".$

Please use the $y$ and $z$ values assigned to you for the following questions.

$(1)$ Assume the time step size $=0\mathrm{.}6s.$ Use Euler's method to calculate the speed of vehicle

$i$ at $t=5\mathrm{.}6s,t=6\mathrm{.}2s$ and $t=6\mathrm{.}8s,$ respectively. Show the detailed calculation process.

$(2)$ Based on Euler's method $($with $=0\mathrm{.}6s),$ use any tool $($C$/$C$++,$ Matlab, Python, Excel,

etc.$)$ to calculate the speed of both vehicles, and the spacing between the two vehicles, from

$t=5s$ to $t=11s.$ Fill the results into the following table, and plot graphs to visualize how

the speeds of the two vehicles and the spacing between the two vehicles change with time

$($from $t=5s$ to $t=11s$

$(3)$ Consider using different $$ values in Question $(2).$ Visualize how speeds and spacing

change with time, from $t=5s$ to $t=11s,$ under different values of $$:$=0\mathrm{.}3s,0\mathrm{.}1s,$ and

$0\mathrm{.}05$ s $.$

$(4)$ Comment on the graphs plotted in Question $(2)$ and Question $(3).$