Result

Relative velocity

v_{A C} = v_{A B} + v_{B C}

Statement

The velocity of A relative to C equals the velocity of A relative to B plus the velocity of B relative to C.

Symbols and units

$v_{A C}$	velocity of A relative to C	m/s
$v_{A B}, v_{B C}$	intermediate relative velocities	m/s

When it applies

Reference frames moving at constant velocity relative to one another (no rotation, no acceleration).

Don't use when

Breaks down for an accelerating or rotating reference frame, where fictitious forces appear and simple vector subtraction no longer holds.

Derivation

Differentiate the position relation r_{AC} = r_{AB} + r_{BC} with respect to time; each term becomes a velocity:

v_{A C} = v_{A B} + v_{B C}

Worked examples

Boat crossing a river

Boat heads straight across at 3 m/s; current runs at 2 m/s downstream:

v_{boat, ground} = (3, - 2) m/s

Ground speed is √13 ≈ 3.6 m/s, angled downstream — pointing the boat straight across is not enough to arrive directly opposite.

Watch out for

Valid only for reference frames moving at constant relative velocity — an accelerating frame introduces additional, fictitious terms.

Connects to