Relative velocity

Notation: $\vec{v}_{A/B}$ reads as the velocity of A relative to B. The rule is a subtraction, $\vec{v}_{A/B} = \vec{v}_A - \vec{v}_B$, with both velocities measured in one shared frame, usually the ground.

One dimension, one sign convention. Take east as positive. Car A travels at $+30$ m/s and car B at $+20$ m/s on the same road, so $v_{A/B} = 30 - 20 = +10$ m/s and A pulls away from B at 10 m/s. Turn B around to $-20$ m/s and $v_{A/B} = 30 - (-20) = +50$ m/s, which is the 50 m/s closing speed quoted for a head-on approach.

Swapping the subscripts flips the sign and changes nothing else: $\vec{v}_{B/A} = -\vec{v}_{A/B}$, so B falls behind at 10 m/s in the first case.

Subtract, do not add. EK 1.4.B.2 says the observed velocity comes from combining the object's velocity with the velocity of the observer's frame, and combining means adding or subtracting vectors, so the observer's direction decides which. Reaching for $v_A + v_B$ by reflex is the standard error.

An AP Physics 1 boundary statement restricts this vector work to one dimension, so signs carry all of the direction. Acceleration survives the change of observer untouched, since every inertial observer measures the same value. Topic 1.4 Reference Frames and Relative Motion covers the AP treatment.