Fundamentals of Vehicle Kinematics in 2D
Kinematics describes the motion of vehicles without considering forces, focusing on position, velocity, and acceleration. In 2D space, we model vehicle motion on a flat plane using coordinates like $x$ (forward) and $y$ (lateral).
Position is the vehicle's location at time $t$, denoted as $\mathbf{r}(t) = (x(t), y(t))$. For example, a car starting at $(0, 0)$ moving straight at constant speed reaches $(10, 0)$ meters after 2 seconds.
Velocity is the rate of change of position: $\mathbf{v}(t) = \frac{d\mathbf{r}}{dt} = (\dot{x}(t), \dot{y}(t))$. Average velocity might be $\mathbf{v}_{avg} = \frac{\Delta \mathbf{r}}{\Delta t}$. In self-driving cars, velocity helps predict if a vehicle will stop at a light.
Acceleration is the rate of change of velocity: $\mathbf{a}(t) = \frac{d\mathbf{v}}{dt} = (\ddot{x}(t), \ddot{y}(t))$. For a car accelerating from 0 to 60 km/h in 5 seconds, $a = \frac{\Delta v}{\Delta t} \approx 3.33$ m/s².
Display math for linear motion: $$x(t) = x_0 + v_0 t + \frac{1}{2} a t^2$$
This equation models a self-driving car merging onto a highway.