Tracking Actuator Angular Velocity (Robotics & Computer Engineering)

A civil defense radar installation uses an automated tracking dish mounted at ground level to monitor oncoming flight vectors. The radar sits at a fixed perpendicular distance of $x = 4\text{ km}$ from the approach flight path. An incoming target flies at a constant altitude of $y = 3\text{ km}$ and approaches the station horizontally with distance coordinate $z$ (in $\text{km}$).

The tracking angle $\theta$ (in radians) of the servo-actuator dish adjusts relative to the plane's horizontal distance coordinate $z$ according to the inverse trigonometric relationship:

As the plane flies inward, its coordinate distance decreases over time $t$ (in seconds) according to the linear tracking velocity:

1. Apply the Inverse Tangent Chain Rule to find the spatial rate of change of the angle with respect to distance ($\frac{d\theta}{dz}$).
2. Simplify $\frac{d\theta}{dz}$ algebraically by clearing compound fractions.
3. Link spatial angular change to temporal tracking speed using the Chain Rule ($\frac{d\theta}{dt} = \frac{d\theta}{dz} \cdot \frac{dz}{dt}$).
4. Calculate the required angular sweep velocity of the servo motor ($\frac{d\theta}{dt}$) when the horizontal distance settles at $z = 3\text{ km}$.