AHL 3.12 (HL)—Vector kinematics

A vector kinematics model writes position as r(t)=r₀+vt for constant velocity, or uses the corresponding component equations when acceleration is present.

To test whether two moving objects meet, solve their position vectors for a common time in the allowed interval. A closest approach is different: minimise the squared separation rather than forcing an exact intersection.

If r₁=(0,0)+t(3,1) and r₂=(6,4)+t(−1,−1), solving gives t=2 and position (6,2); the same time in both equations is the evidence of an intersection.

Equal coordinates at different times do not mean collision. State the time domain and distinguish position from velocity.

For variable velocity in two dimensions, $v(t)=r'(t)$ and $r(t)=\int v(t)\,dt+C$, with the initial position fixing $C$; speed is $|v(t)|$. Relative position of $B$ from $A$ is $r_B-r_A$. To find closest approach, minimize $|r_B-r_A|^2$ over the allowed time interval and check endpoints as well as stationary values; projectile and circular motion are special cases of this vector model.