Aerospace Engineering (Satellite Orbit Path Modeling) Solution

Problem Statement

A communication satellite passes through initial orbital checkpoint $A(4200, 1500, 6300)$ moving along a linear trajectory toward deep-space relay node $B(4350, 1620, 6500)$ (coordinates in kilometers).

Step-by-Step Mathematical Solution

1. Vector Equation of the Trajectory

Compute the direction displacement vector $\vec{v} = \vec{AB}$ :

\vec{v} = \langle 4350 - 4200, 1620 - 1500, 6500 - 6300 \rangle = \langle 150, 120, 200 \rangle\text{ km}

Using checkpoint $A$ as initial position vector $\vec{r}_0 = \langle 4200, 1500, 6300 \rangle$ , the vector equation is:

\vec{r}(t) = \langle 4200, 1500, 6300 \rangle + t\langle 150, 120, 200 \rangle = \langle 4200 + 150t, 1500 + 120t, 6300 + 200t \rangle

2. Parametric Equations

Separating into scalar Cartesian coordinates:

$x(t) = 4200 + 150t$
$y(t) = 1500 + 120t$
$z(t) = 6300 + 200t$

3. Coordinate Position at $t = 0.4$

Substitute $t = 0.4$ into the parametric equations:

$x(0.4) = 4200 + 150(0.4) = 4200 + 60 = 4260\text{ km}$
$y(0.4) = 1500 + 120(0.4) = 1500 + 48 = 1548\text{ km}$
$z(0.4) = 6300 + 200(0.4) = 6300 + 80 = 6380\text{ km}$

The exact coordinates of the satellite at $t = 0.4$ are $(4260, 1548, 6380)\text{ km}$ .

Real-Life Engineering Meaning & Application

Physical Interpretation: The parameter $t$ acts as a normalized mission progress scalar where $t=0$ corresponds to insertion at checkpoint $A$ and $t=1$ corresponds to arrival at relay node $B$ . The evaluated coordinate $(4260, 1548, 6380)$ represents the exact spatial pinpoint of the spacecraft at $40\%$ of the scheduled cruise leg.
Engineering Application: Ground-based telemetry tracking stations at UNLV and NASA Deep Space Network facilities continuously align parabolic high-gain antennas using these exact parametric coordinates. If the observed radar return deviates from the predicted $(x, y, z)$ waypoint, guidance algorithms automatically calculate delta-V thruster burn vectors to correct the trajectory.