Relative Velocity Formula: v_AB = v_A − v

The relative velocity formula is $\vec{v}_{AB} = \vec{v}_A - \vec{v}_B$ . It gives the velocity of object A as it would be measured by an observer riding along with object B. Subtract B's velocity from A's, as vectors, and you have the motion of A seen from B.

Velocity always depends on who is watching. A passenger walking down a moving train at 1 m/s is doing 1 m/s to fellow passengers but much more to someone on the platform. In one dimension the formula is plain subtraction with signs. In two dimensions, such as a boat crossing a river or rain falling on a moving cyclist, the subtraction is done with vector components or a right angled triangle.

Quick facts

Formula
$\vec{v}_{AB} = \vec{v}_A - \vec{v}_B$ (A seen from B)
Same direction
$v_{AB} = v_A - v_B$
Opposite directions
$v_{AB} = v_A + v_B$ in size
Reverse view
$\vec{v}_{BA} = -\vec{v}_{AB}$
SI unit
m/s
Type
Vector

Relative velocity formula in one dimension

Choose one direction as positive and give every velocity its sign. Then subtract. Two cases cover most questions:

Relative speed of two bodies on a straight line
Motion	Relative speed	Example
Same direction	$\|v_A - v_B\|$	25 m/s and 18 m/s give 7 m/s
Opposite directions	$v_A + v_B$	15 m/s and 20 m/s give 35 m/s
One at rest	$v_A$	Ground view, the usual case

Notice that A relative to B and B relative to A have the same size and opposite directions. If you see a car ahead pulling away at 7 m/s, the driver of that car sees you falling back at 7 m/s.

Worked examples

Example 1: overtaking on a highway

On a straight highway, car A cruises at 25 m/s while car B, ahead of it in the same lane direction, does 18 m/s. What is the velocity of A relative to B?

$v_{AB} = 25 - 18 = 7\ \text{m/s}$

From inside B, car A appears to creep forward at 7 m/s.

Example 2: two trains passing, converting km/h

A 150 m train at 54 km/h meets a 200 m train at 72 km/h on parallel tracks, moving the opposite way. How long do they take to pass each other completely?

Speeds: 54 ÷ 3.6 = 15 m/s and 72 ÷ 3.6 = 20 m/s. Opposite directions, so the relative speed is 15 + 20 = 35 m/s.

Distance to clear = 150 + 200 = 350 m.

$t = \frac{350}{35} = 10\ \text{s}$

Example 3: a boat crossing a river

A boat heads straight across an 80 m wide river at 4.0 m/s through the water. The current flows at 3.0 m/s. Find the boat's velocity relative to the bank, the crossing time and how far downstream it lands.

Velocity over the bank: $\sqrt{4.0^2 + 3.0^2} = 5.0$ m/s, at $\tan^{-1}(3/4) \approx 37^\circ$ downstream of the straight across line.

Crossing time = 80 ÷ 4.0 = 20 s. Drift = 3.0 × 20 = 60 m downstream.

Going from the moving frame back to the ground

Rearranging the formula gives $\vec{v}_A = \vec{v}_{AB} + \vec{v}_B$ : the ground velocity of A is its velocity relative to B plus B's own ground velocity. A passenger walks toward the front of a train at 1.5 m/s while the train moves at 20 m/s. To someone on the platform the passenger moves at 1.5 + 20 = 21.5 m/s. If the passenger turns and walks toward the back, the platform view is 20 − 1.5 = 18.5 m/s, still forward.

The same rule handles aircraft in wind and swimmers in a current. Add the velocity through the medium to the velocity of the medium itself.

Two dimensional relative velocity

When the velocities are not along one line, write each as components, subtract the x parts and the y parts separately, and recombine with Pythagoras. Rain is a common exam case. Rain falling straight down at 6 m/s looks slanted to a cyclist riding at 8 m/s. Relative to the cyclist, the rain has 6 m/s downward and 8 m/s backward, giving 10 m/s. The cyclist should tilt the umbrella forward, at an angle whose tangent is 8/6 to the vertical, about 53°.

Mistakes that cost marks

Wrong order. v_AB means A seen from B, so it is v_A − v_B, not v_B − v_A.
Ignoring signs for opposite directions. Subtracting a negative velocity adds the speeds.
Adding train lengths wrongly. To pass completely, the trains clear the sum of their lengths.
Using the drift speed for crossing time. Only the velocity component across the river decides how long the crossing takes.

The vector basics are on the velocity formula page, and changes of velocity on the change in velocity formula page. Every other motion topic is indexed under kinematics formulas.

Frequently asked questions

What is the formula of relative velocity?

Write v_AB = v_A − v_B. This gives the velocity of A as measured by an observer moving with B. Subtract the velocities as vectors.

What is the relative velocity of two bodies moving in opposite directions?

Its size is the sum of their speeds, v_A + v_B, because one velocity is negative relative to the other.

Is relative velocity a vector quantity?

It is a vector. Swapping the observer flips the arrow: v_BA always equals minus v_AB, with the same magnitude.

Why does a river current not change the crossing time?

If the boat points straight across, only its velocity across the river carries it to the far bank. The current only carries it downstream.

What is the relative velocity of two cars moving at the same velocity?

Zero. Each driver sees the other at rest, even though both move relative to the road.