Relative Velocity

What this formula says

Every velocity is measured relative to some observer. When we say a car is moving at $60$ km/h, we mean relative to the ground. But an observer in another moving car sees a different velocity.

The velocity of object A as seen by observer B is:

$\vec{v}_{AB} = \vec{v}_A - \vec{v}_B$

Here:

• $\vec{v}_A$ is the velocity of A relative to the ground
• $\vec{v}_B$ is the velocity of B relative to the ground
• $\vec{v}_{AB}$ is the velocity of A as seen by B (A relative to B)

Building the concept from scratch

Imagine you are standing on the ground watching two cars on a straight road.

• Car A moves at $+60$ km/h (to the right)
• Car B moves at $+40$ km/h (to the right)

From the ground, both are moving right. But a passenger in car B sees car A moving away — but only at $60 - 40 = 20$ km/h. Car A appears slow from car B's perspective.

Now imagine car B is moving at $-40$ km/h (to the left). The passenger in car B now sees car A approaching at $60 - (-40) = 100$ km/h — much faster, because they are moving toward each other.

This is the physical meaning of $\vec{v}_{AB} = \vec{v}_A - \vec{v}_B$.

Derivation

Let $x_A$ and $x_B$ be the positions of A and B at time $t$, measured from a fixed point on the ground.

The position of A relative to B is:

$x_{AB} = x_A - x_B$

Differentiate both sides with respect to time:

$\frac{d x_{AB}}{dt} = \frac{d x_A}{dt} - \frac{d x_B}{dt}$

By definition, $\frac{dx}{dt}$ is velocity. So:

$\boxed{\vec{v}_{AB} = \vec{v}_A - \vec{v}_B}$

The formula is a direct consequence of subtracting positions and differentiating. Nothing more is assumed.

Direction matters — this is a vector equation

In one dimension, the sign carries the direction. In two or three dimensions, the velocities are full vectors and must be subtracted as vectors — including direction.

Example in 1D:

Two trains on parallel tracks:

• Train A: $80$ km/h east → $\vec{v}_A = +80$
• Train B: $50$ km/h east → $\vec{v}_B = +50$

Velocity of A relative to B: $\vec{v}_{AB} = 80 - 50 = +30 \text{ km/h east}$

A passenger in train B sees train A moving east at $30$ km/h — slowly pulling ahead.

Same trains, but B moving west:

• Train B: $50$ km/h west → $\vec{v}_B = -50$

$\vec{v}_{AB} = 80 - (-50) = +130 \text{ km/h east}$

The trains are approaching each other — from B's perspective, A is rushing toward it at $130$ km/h.

The reverse: velocity of B relative to A

$\vec{v}_{BA} = \vec{v}_B - \vec{v}_A = -\vec{v}_{AB}$

The relative velocity reverses when you swap who is observing whom. If A sees B moving at $+30$ km/h east, then B sees A moving at $-30$ km/h east (i.e. $30$ km/h west). This is always true.

Relative velocity in two dimensions

When the two objects move in different directions, subtract the velocity vectors component by component.

Object A moves at $v_A$ at angle $\alpha$ from the x-axis. Object B moves at $v_B$ at angle $\beta$ from the x-axis.

Components of relative velocity:

$(\vec{v}_{AB})_x = v_A\cos\alpha - v_B\cos\beta$

$(\vec{v}_{AB})_y = v_A\sin\alpha - v_B\sin\beta$

Magnitude of relative velocity:

$|\vec{v}_{AB}| = \sqrt{(\vec{v}_{AB})_x^2 + (\vec{v}_{AB})_y^2}$

Special cases:

• Moving in the same direction: $|\vec{v}_{AB}| = |v_A - v_B|$ — relative velocity is the difference
• Moving in opposite directions: $|\vec{v}_{AB}| = v_A + v_B$ — relative velocity is the sum
• Moving at right angles: $|\vec{v}_{AB}| = \sqrt{v_A^2 + v_B^2}$