Automated High-Speed Rail Jerk Analysis (Robotics & Computer Engineering)

A computer systems robotics engineer is programming the trajectory control computer for an autonomous airport people mover. The forward position $s(t)$ of the passenger vehicle along a linear guideway (in meters, for $t \in [0, 10]$ seconds) is programmed as:

In vehicular dynamic control, passenger ride comfort and mechanical powertrain stress are governed by higher-order derivatives:

• Velocity: $v(t) = s'(t) = \frac{ds}{dt}$
• Acceleration: $a(t) = s''(t) = \frac{d^2s}{dt^2}$
• Jerk: $j(t) = s'''(t) = \frac{d^3s}{dt^3}$

1. Compute the analytical formulas for velocity $v(t)$, acceleration $a(t)$, and jerk $j(t)$ using successive power rule differentiation.
2. Determine the vehicle's instantaneous jerk at $t = 2\text{ s}$ and $t = 6\text{ s}$.
3. International passenger transit standards require jerk to remain bounded within $|j(t)| \le 1.5\text{ m/s}^3$. Verify whether this vehicle profile satisfies safety standards over the entire $10\text{-second}$ acceleration window.