Radar-Guided Missile Intercept Tracking (Aerospace Engineering)

At the Nevada National Security Site defense test range near Nellis Air Force Base, an automated ground-based phased-array radar installation tracks an incoming supersonic target drone. The drone cruises along a straight, horizontal flight vector at a constant altitude of $y = 8.0\text{ km}$ directly toward the radar station.

The radar transmitter emits high-frequency radio pulses that reflect off the drone, measuring the direct line-of-sight distance (slant range $z$) and the instantaneous range closure rate $\frac{dz}{dt}$ via Doppler frequency shift.

At the precise moment when the radar instruments record a slant range of $z = 10.0\text{ km}$, the tracking telemetry indicates the slant range is decreasing at a rate of:

1. Establish the Pythagorean geometric constraint relating the horizontal ground position $x$, the constant altitude $y$, and the slant range $z$. Apply implicit differentiation with respect to time $t$ to derive the equation governing horizontal closure velocity $\frac{dx}{dt}$.
2. Determine the snapshot horizontal ground distance $x$ from the radar station to the drone, and compute the drone's true horizontal ground velocity $\frac{dx}{dt}$ in $\text{km/s}$ and $\text{km/h}$.
3. Assuming the speed of sound in the upper atmosphere is $a = 300\text{ m/s}$, calculate the target's flight Mach number ($M = v/a$), and explain why aerospace target tracking microcontrollers must apply this related rates decomposition in real time.