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:

\vec{r}(t) = \langle 6\sin t, \; 6(1 - \cos t), \; 8t \rangle\text{ meters for } t \in [0, 2\pi]

where

t

is in seconds.

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

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