Automotive Cornering & Tire Traction Limits | Tony's Intertwingled Memex

In automotive and autonomous vehicle engineering, electronic stability control (ESC) systems continuously monitor vehicle kinematics to prevent skidding. By decomposing the total acceleration vector into its tangential component $a_T$ (longitudinal drive torque/braking) and normal component $a_N$ (lateral cornering grip), vehicle computers compare lateral acceleration against the friction limit $a_{\text{grip}} = \mu_s g$ of the tire contact patch.

An autonomous electric vehicle prototype is undergoing track testing at the Las Vegas Motor Speedway. Its 2D trajectory across the asphalt is tracked in meters as a function of time $t$ (in seconds) by high-precision RTK-GPS:

\vec{r}(t) = \langle t^3 - 3t, 2t^2, 0 \rangle

The tires provide a maximum lateral (normal) acceleration threshold of

a_{N,\max} = 12\text{ m/s}^2

before static friction breaks and the vehicle enters an uncontrolled lateral slide.

State the formulas for the tangential component $a_T$ and normal component $a_N$ of acceleration.
Compute the exact velocity $\vec{v}(2)$ , acceleration $\vec{a}(2)$ , and speed $\|\vec{v}(2)\|$ at time $t = 2\text{ seconds}$ .
Calculate the tangential acceleration $a_T$ and normal acceleration $a_N$ at $t = 2\text{ seconds}$ .
Determine if the vehicle maintains tire traction or if it exceeds the critical skid limit at $t = 2\text{ seconds}$ .