Relative Motion System Analysis

System of Equations Description

The system describes the relative motion of two objects A and B in 3D space:

First equation: Describes the y-coordinate relationship
Second equation: Describes the z-coordinate relationship
Third equation: Velocity magnitude constraint for object B

\[ \begin{cases} y_a + v_{ay}\frac{x_b - x_a}{v_{ax} - v_{bx}} = y_b + v_{by}\frac{x_b - x_a}{v_{ax} - v_{bx}} \\[1em] z_a + v_{az}\frac{x_b - x_a}{v_{ax} - v_{bx}} = z_b + v_{bz}\frac{x_b - x_a}{v_{ax} - v_{bx}} \\[1em] v_{bx}^2 + v_{by}^2 + v_{bz}^2 = s_b^2 \end{cases} \]

System of Equations

Variables

Solution Method

Variables Definition

(x_a, y_a, z_a): Position of object A
(x_b, y_b, z_b): Position of object B
(v_ax, v_ay, v_az): Velocity components of object A
(v_bx, v_by, v_bz): Velocity components of object B
s_b: Speed magnitude constraint for object B