Two vehicles, numbered i and $i-1,$ are travelling on a single$-$lane freeway. At time $t=5s,$ both vehicles have a speed of $15\frac{m}{s},$ and vehicle $i$ is behind vehicle $i-1$ with a clearance of $70$ meters. Vehicle $i$ is $4$ meters long, and vehicle $i-1$ is $4\mathrm{.}5$ meters long. Assume:

For vehicle $i-1$ : during $t=5s$ and $t=7s,$ it brakes with a constant deceleration of $-3\frac{m}{s^2}.$ After $t=7s,$ 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 $m=2,n=0\mathrm{.}5,$ and ${}_5=0\mathrm{.}6sec*$ meter ${?}^{-1\mathrm{.}5}$

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

$(1)$ Assume $t=0\mathrm{.}2s.$ Use Euler's method to calculate the speed of vehicle $i$ at $t=5\mathrm{.}2s,t=5\mathrm{.}4s$ and $t=5\mathrm{.}6s,$ respectively. Show the detailed calculation process.

$(2)$ Based on Euler's method $($with $t=0\mathrm{.}2s,$ use any tool $(\frac{C}{C}++,$ Matlab, Python, Excel, etc.$)$ to calculate the speed of both vehicles, and the spacing between the two vehicles, from $t=5s$ to $t=10s.$ Present the code $($or the screen capture of the tool if programming language is not used$),$ show the results in tables, and plot graphs to visualize how speeds and spacing changes with time.

$(3)$ Consider using different $t$ values in Question $(2).$ Visualize how speeds and spacing change with time, from $t=5s$ to $t=10s,$ under different values of $t$:$t=0\mathrm{.}05s,0\mathrm{.}1s,0\mathrm{.}2s$ and $0\mathrm{.}5s.$

$(4)$ Comment on the results in Question $(3).$