Relative Velocity in One Dimension

Every velocity we have discussed so far has implicitly been measured with respect to the ground (our chosen inertial frame, Section 2.1). But velocity is always relative to the observer's frame, and it is often useful — especially when two or more bodies move along the same straight line — to know how fast one body appears to move as seen from another moving body.

Definition. If two objects A and B move along the same straight line with velocities $v_A$ and $v_B$ (measured with respect to the ground, and with the usual sign convention for direction), the velocity of A relative to B is defined as

and, symmetrically, the velocity of B relative to A is $v_{BA} = v_B - v_A = -v_{AB}$. Relative velocity along a straight line is obtained by simple algebraic (signed) subtraction, precisely because both velocities lie along the same one-dimensional axis.

Case 1 — bodies moving in the same direction. If A and B move in the same direction with speeds $|v_A|$ and $|v_B|$, the magnitude of their relative velocity is simply the difference of their speeds, $||v_A| - |v_B||$. For example, if car A moves at 20 m/s and car B moves at 15 m/s in the same direction, A moves away from B (or gains on B, if it started behind) at a relative velocity of $20 - 15 = 5$ m/s; equivalently, B falls behind A at 5 m/s, i.e. B's velocity relative to A is $-5$ m/s.

Case 2 — bodies moving in opposite directions. If A and B move toward each other (or away from each other) along the same line, their velocities have opposite signs once a common positive direction is fixed, and the magnitude of the relative velocity is the sum of their speeds. For example, if car A moves at 20 m/s in the positive direction and car B moves at 15 m/s in the negative direction (i.e. $v_B = -15$ m/s), the velocity of A relative to B is $v_A - v_B = 20 - (-15) = 35$ m/s — meaning the two cars approach (or separate from) each other at a combined rate of 35 m/s, considerably faster than either car's individual speed relative to the ground.

Special case — a body at rest. If B is stationary ($v_B = 0$), then $v_{AB} = v_A$: relative velocity reduces to ordinary velocity, as it must, since a stationary body shares the ground's own inertial frame. …