Relative velocity and acceleration

Relative velocity

If two particles A and B have velocities $\vec v_A$ and $\vec v_B$ (both measured in a fixed frame), the relative velocity of A with respect to B is the velocity of A as seen by an observer moving with B:

$$\vec v_{A/B}=\vec v_A-\vec v_B$$

Equivalently, $\vec v_A=\vec v_B+\vec v_{A/B}$. It follows from the relative position $\vec r_{A/B}=\vec r_A-\vec r_B$ by differentiating.

Relative acceleration

In the same way, the relative acceleration of A with respect to B is

$$\vec a_{A/B}=\vec a_A-\vec a_B$$

(valid when the observer on B translates without rotating). Note that $\vec v_{B/A}=-\vec v_{A/B}$ and $\vec a_{B/A}=-\vec a_{A/B}$.

Example

Car A moves east at 60 km/h and car B moves north at 80 km/h. The velocity of A relative to B is

$$\vec v_{A/B}=60\hat i-80\hat j\ \text{km/h},\qquad |v_{A/B}|=\sqrt{60^2+80^2}=100\ \text{km/h}$$

at $\tan^{-1}(80/60)=53.1^\circ$ south of east. A passenger in B sees A moving at 100 km/h in that direction. If A accelerates at $2\hat i$ m/s$^2$ and B at $1\hat j$ m/s$^2$, then $\vec a_{A/B}=2\hat i-1\hat j$ m/s$^2$.