Relative Velocity in One and Two Dimensional Motion

When two objects A and B move with their own velocities, what an observer riding along with B actually perceives is not A's velocity by itself, but the relative velocity of A with respect to B — the rate at which A's position changes as seen from B. It is defined as the vector difference of the two velocities:

Case 1 — same direction. If A and B move along parallel straight tracks in the same direction with speeds $v_A$ and $v_B$ (measured with respect to the ground), then $v_{AB} = v_A - v_B$ and $v_{BA} = v_B - v_A$: the magnitude of the relative velocity of one with respect to the other equals the difference of their speeds. (E.g. car A at 35 km h$^{-1}$ east and car B at 40 km h$^{-1}$ east: to a passenger in A, car B appears to creep ahead eastward at 5 km h$^{-1}$; to a passenger in B, car A appears to fall behind, i.e. drift westward, at 5 km h$^{-1}$.)

Case 2 — opposite directions. If A and B move along the same track but in opposite directions, then their relative velocity is $v_{AB}=v_A-(-v_B)=v_A+v_B$: the magnitude of the relative velocity equals the sum of their speeds.

Case 3 — general angle $\theta$ between $\vec v_A$ and $\vec v_B$. Using the law-of-cosines result from vector subtraction (§2.3.4),

(with $\beta$ the angle between $\vec v_{AB}$ and $\vec v_B$). This reduces to Case 1 when $\theta=0°$ ($v_{AB}=v_A-v_B$, along $\vec v_A$; and correspondingly $v_{BA}=v_B+v_A$ along $\vec v_B$) and to Case 2 when $\theta=180°$ ($v_{AB}=v_A+v_B$, along $\vec v_A$; $v_{BA}=v_B-v_A$ along $\vec v_B$). For two bodies moving at right angles ($\theta=90°$), $v_{AB}=\sqrt{v_A^2+v_B^2}$. …