Relative Velocity in a Plane

When two objects are both moving, it is often useful to ask: "how does the motion of one

object appear to an observer riding along with the other?" The answer is the relative velocity, and — because velocity is a vector — it must be computed by vector subtraction,

never by simply subtracting speeds (which only works when both velocities happen to lie

along the same straight line).

If $\vec v_A$ and $\vec v_B$ are the velocities of objects $A$ and $B$ measured with respect

to a common, fixed reference frame (usually the ground), the velocity of $A$ relative to

$B$ is defined as

Since this is vector subtraction, it is carried out either graphically (triangle law with the

negative of $\vec v_B$) or, far more conveniently, by the component method of Section 3.7:

subtracting the x-components and y-components of $\vec v_A$ and $\vec v_B$ separately, then

recombining to get the magnitude and direction of $\vec v_{AB}$.

Two classic situations illustrate this idea. In a river-crossing problem, a boat's

velocity relative to the ground is the vector sum of the boat's velocity relative to the

water (set by its engine/rowing, in whatever direction it is steered) and the water's own

velocity relative to the ground (the current) — the two velocities add as vectors, not as

speeds, because they generally point in different directions. In a rain-and-walker problem, rain falling vertically appears, to a person walking or running horizontally, to

come from a slanted direction ahead of them; the apparent (relative) velocity of the rain as …