Velocity

Prof. Dr. Thorsten Jungmann (last updated 2025-09-03)

Velocity (German: Geschwindigkeit) describes the change in position per unit of time. It is actually a vector quantity, but in the context of homogeneous fields and rectilinear motion it is treated as a scalar quantity.

Vector quantity in a homogeneous field → scalar quantity

This is a vector quantity. In this script, however, it is treated by convention as a scalar quantity in the context of the homogeneous field (for a detailed justification).

1 Quantity symbol and unit

The quantity symbol is $\vec{v}$ or $v$ (as a scalar quantity).
The unit is metres per second ($m/s$).
Velocity is defined as the derivative of distance with respect to time:

$$v(t)=\dot{s}(t)=\frac{\mathrm{d}s}{\mathrm{d}t}=\underset{\Delta t\to 0}{\lim }\frac{\Delta s}{\Delta t}$$

where $\Delta s$ is the distance travelled and $\Delta t$ is the time required for it.

2 Practical and worked examples

• The speed of light in a vacuum is $c=299792458m/s\approx 3\cdot 10^{8}m/s$.
• The drift velocity of electrons in copper conductors is about $0.1mm/s$.
• The speed of sound in air is about $343m/s$ at $20°\mathrm{C}$.
• Electromagnetic waves in cables reach about $60\%$ to $90\%$ of the speed of light.

3 Typical orders of magnitude

• Drift velocity of electrons: $10^{-5}$ to $10^{-3}m/s$
• Speed of sound: $10^2$ to $10^{3}m/s$
• Electromagnetic waves: $10^{8}m/s$ (order of magnitude of the speed of light)
• Particles in accelerators: up to $0.99\cdot c$

4 Example phrasings

• ‘The propagation velocity of the electromagnetic wave is $2.1\cdot 10^{8}m/s$.’
• ‘At a voltage of $230\mathrm{V}$, electrons reach a mean drift velocity of $0.1mm/s$.’
• ‘Signals in data cables propagate at about $200000km/s$.’

5 Related quantities

• Length: $s=v\cdot t$ at constant velocity
• Time: $t=\frac{s}{v}$
• Force: For moving charges in a magnetic field: $F=q\cdot v\cdot B$
• Frequency: For waves: $v=\lambda \cdot f$
• Wavelength: $\lambda =\frac{v}{f}$

6 Relevant components

• Conductor
• Capacitor
• Inductor
• Transformer