Relative velocity

In non-relativistic kinematics, relative velocity is the vector difference between the velocities of two objects, as evaluated in terms of a single coordinate system.

For example, if the velocities of particles A and B are $\mathbf{v}_A$ and $\mathbf{v}_B$ respectively in terms of a given coordinate system, then the relative velocity of A with respect to B (also called the velocity of A relative to B, $\mathbf{v}_{A/B}$, or $\mathbf{v}_{A \mathrm{\ rel\ } B}$) is

$\mathbf{v}_{A \mathrm{\ rel\ } B} = \mathbf{v}_A - \mathbf{v}_B$.

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".

Conversely, the velocity of B relative to A is

$\mathbf{v}_{B \mathrm{\ rel\ } A} = \mathbf{v}_B - \mathbf{v}_A$.

To be clear, the velocity of B "as seen from A" (which is to say, $\mathbf{v}_B - \mathbf{v}_A$) is another expression meaning "the velocity of B relative to A" or indeed "the velocity of B in the coordinate system where A is considered at rest"

Example

Joe and Sara are driving in the same direction. Joe’s velocity is 90 km/h and Sara’s 100 km/h. If we take Joe’s velocity as $\mathbf{v}_B$ and Sara’s $\mathbf{v}_A$ then

$\mathbf{v}_{A \mathrm{\ rel\ } B} = 100 - 90 = 10 \text{ km/h.}$

This is the velocity observed by Joe. Joe sees Sara moving at 10 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

External links

• Relative Motion at HyperPhysics
• A Java applet illustrating Relative Velocity, by Andrew Duffy