# Relative velocity

The relative velocity $\vec{v}_\mathrm{BA}$ (also $\vec{v}_\mathrm{B/A}$ or $\vec{v}_\mathrm{B rel A}$) is the velocity of an object or observer B in the rest frame of another object or observer A, if it is constant,

$\vec{v}_\mathrm{BA}=-\vec{v}_\mathrm{AB}$

where $\vec{v}_\mathrm{AB}$ is A's velocity in the rest frame of B.

## Classical mechanics

Relative velocities between two particles in classical mechanics.

In the Newtonian limit where approximately the Galilean transformation

$\vec{r}'=\vec{r}-\vec{v}t$
$t'=t$

holds, it is identical with the vector difference between the velocities of A and B, as evaluated in terms of a single coordinate system:

$\vec{v}_\mathrm{BA}=\vec{v}_\mathrm{B}-\vec{v}_\mathrm{A}$.

Due to Einstein's Special Relativity (SR), doesn't hold in general. The expression "the velocity of A relative to B" is shorthand for "the velocity of A in the coordinate system where B is always at rest".

Suppose there is a reference frame A rotating with an angular velocity vector ω, and having a translational velocity vector of V with respect to the ground, and there is a different body B whose translational velocity vector is U with respect to A, then the velocity vector of B with respect to the ground is given as

    Ú = V + U(considering ω = 0) + (ω x R)


where R is the position vector of B with respect to A.

Acceleration vector of B with respect to ground ...

  a = d/dt(V) + {d/dt(U) (considering ω =0)} + R x d/dt(ω) + ω x (ω x R) + 2ω x U


## Special relativity theory

According to SR, the vacuum speed of light is isotropically equal to a universal constant c in any coordinate systems. Inter alia, it leads to the fact that

• apart from Newtonian limit, velocities are not additive quantities, and
• the difference velocity between A and B is not equal to their relative velocity and particularly has a smaller absolute value. Whereas the maximum difference speed between two objects is 2c, the maximum absolute value of a relative velocity is equal to c.

To get $\vec{v}_\mathrm{BA}$ from $\vec{v}_\mathrm{A}$ and $\vec{v}_\mathrm{B}$ in an arbitrary reference frame, it's necessary to Lorentz transformation the latter into the rest frame of A. If $\vec{v}_\mathrm{A}$ and $\vec{v}_\mathrm{B}$ are collinear, the formula

$v_\mathrm{BA}=\frac{v_\mathrm{B}-v_\mathrm{A}}{1-\frac{v_\mathrm{A}v_\mathrm{B}}{c^2}}$

holds.

## Example

Joe and Sara are driving in the same direction. Joe’s velocity is 90 km/h and Sara’s 110 km/h. If we take Joe’s velocity as $\vec{v}_B$ and Sara’s $\vec{v}_A$ then

$\vec{v}_{A \mathrm{\ rel\ } B} = 110 - 90 = 20 \text{ km/h.}$

This is the velocity observed by Joe. Joe sees Sara moving at 20 km/h.

Joe and Sara are driving in the opposite directions i.e. heading towards each other or moving away from each other. Joe’s velocity is 90 km/h and Sara’s 100 km/h. If we take Joe’s velocity as $\vec{v}_B$ and Sara’s $\vec{v}_A$ then

$\vec{v}_{A \mathrm{\ rel\ } B} = 100 - (-90) = 190 \text{ km/h.}$

Note that in this case Joe's velocity is negative, as its direction is opposite Sara's velocity. Therefore...

$\vec{v}_{A \mathrm{\ rel\ } B} = 100 + 90 = 190 \text{ km/h.}$

This is the velocity observed by Joe. Joe sees Sara moving at 190 km/h.

## References

• Alonso & Finn, Fundamental University Physics ISBN 10:0-201-56518-8
• Greenwood, Donald T, Principles of Dynamics.
• Goodman and Warner, Dynamics.
• Beer and Johnston, Statics and Dynamics.
• McGraw Hill Dictionary of Physics and Mathematics.
• Rindler, W., Essential Relativity.
• KHURMI R.S., Mechanics, Engineering Mechanics, Statics, Dynamics