Structural Safe-Load Limits for Cargo Cranes (Mechanical Engineering)

At an intermodal container logistics terminal, a heavy-duty gantry crane hoists an industrial cargo container vertically along a vertical guide rail track. The hoist cable is pulled in by a motorized winch drum mounted on a high overhead gantry beam at a fixed horizontal distance of $x = 24.0\text{ meters}$ from the vertical path of the container.

As the winch drum spools the high-strength steel cable inward, crane load sensors monitor the active cable length $z$ (the straight-line distance from the drum to the lifting spreader) and the reel-in velocity $\frac{dz}{dt}$.

At a critical lifting checkpoint, the cable length is $z = 30.0\text{ meters}$, and the motorized winch spools cable inward at a constant linear speed of:

1. Model the crane geometry using a right triangle with fixed horizontal offset $x$, dynamic vertical distance $y$ (measured downward from drum level to the container), and cable hypotenuse $z$. Apply implicit time differentiation to establish the relationship between $\frac{dz}{dt}$ and the vertical container velocity $\frac{dy}{dt}$.
2. Compute the vertical position $y$ of the container at the snapshot moment, and determine the exact instantaneous vertical hoist velocity $\frac{dy}{dt}$ in $\text{m/s}$.
3. Structural safety codes for container hoists mandate that vertical lift speeds must never exceed $3.0\text{ m/s}$ to prevent dynamic line-tension spikes and resonant pendulum oscillations. Evaluate whether the crane operates within safe regulatory limits, and explain why crane control microcontrollers must compute $\frac{dy}{dt}$ rather than relying on winch spool speed $\frac{dz}{dt}$.