We are given a car model: $$\dot x = V\cos(a) \quad \dot y = V\sin(a) \quad \dot a = u$$
$V$ some arbitrary number
Make an (first order) approximation $$\dot x = V \quad \dot y = Va \quad \dot a = u$$
And discretize into: $$(1) \quad a(k+1) = a(k) + u(k)*T$$ $$(2) \quad y(k+1) = y(k) + a(k)VT + u(k)*0.5V*T^2$$
I can understand how they got the first equation, but how was the second equation derived? Specifically, where did that $0.5$ term come from?
(on a side note, is there a name for discretization of the form $\dot a \approxeq \dfrac{a(k+1) - a(k)}{T}?) $
The second piece comes from averaging two different discretizations of $\dot{y}$. Namely, we could replace the $a$ in the $\dot{y}=Va$ equation by either $a(k)$ or $a(k+1)$. This gives us two equations: $$y(k+1)=y(k)+a(k)VT$$ and: $$y(k+1)=y(k)+a(k+1)VT$$ Then averaging the two equations yields: $$\frac{y(k+1)+y(k+1)}{2}=\frac{y(k)+a(k)VT+y(k)+a(k+1)VT}{2}$$ Simplifying this equation and substituting the expression for $a(k+1)$ from equation (1) of your question then gives: $$y(k+1)=y(k)+a(k)VT+\frac{u(k)VT^2}{2}$$
As far as I am aware, the process of replacing $\dot{f}$ by $\frac{f(k+1)-f(k)}{T}$ is referred to as discretization.