Roller Coaster Loop Structural Dynamics (Entertainment & Mechanical Engineering)

An entertainment engineering theme park ride features a high-speed roller coaster vehicle traversing an inverted loop whose 3D spatial path is parameterized by:

To prevent structural track fatigue and ensure passenger comfort within biomechanical G-force limits:

1. Find the velocity $\vec{v}(t)$, acceleration $\vec{a}(t)$, and scalar speed $v(t) = \|\vec{v}(t)\|$.
2. Compute the total track length $s$ traversed during one complete loop from $t = 0$ to $t = 2\pi$.
3. Evaluate the curvature $\kappa(t)$ and radius of curvature $\rho(t)$ along the trajectory.
4. Compute the tangential acceleration $a_T$ (throttle/braking) and normal acceleration $a_N$ (centripetal turning).
5. Verify whether the normal acceleration exceeds the passenger safety limit of $a_{N,\max} = 1.0\text{ g} = 9.81\text{ m/s}^2$.