Summarizing Relativistic Addition of Velocities

Relativistic Addition of Velocities Summary

• With classical velocity addition, velocities add like regular numbers in one-dimensional motion: \(\text{u=v+u}\prime \), where \(v\) is the velocity between two observers, \(u\) is the velocity of an object relative to one observer, and \(u\prime \) is the velocity relative to the other observer.
• Velocities cannot add to be greater than the speed of light. Relativistic velocity addition describes the velocities of an object moving at a relativistic speed:
  \(\text{u=}\cfrac{\text{v+u}\prime }{1+\cfrac{v\text{u}\prime }{{c}^{2}}}\)
• An observer of electromagnetic radiation sees relativistic Doppler effects if the source of the radiation is moving relative to the observer. The wavelength of the radiation is longer (called a red shift) than that emitted by the source when the source moves away from the observer and shorter (called a blue shift) when the source moves toward the observer. The shifted wavelength is described by the equation

  \({\lambda }_{\text{obs}}{\text{=λ}}_{s}\sqrt{\cfrac{1+\cfrac{u}{c}}{1-\cfrac{u}{c}}}\)

  \({\lambda }_{\text{obs}}\) is the observed wavelength, \({\lambda }_{s}\) is the source wavelength, and \(u\) is the relative velocity of the source to the observer.

Glossary

classical velocity addition

the method of adding velocities when \(v\text{<<}c\); velocities add like regular numbers in one-dimensional motion: \(u=\mathrm{v+u}\prime \), where \(v\) is the velocity between two observers, \(u\) is the velocity of an object relative to one observer, and \(u\prime \) is the velocity relative to the other observer

relativistic velocity addition

the method of adding velocities of an object moving at a relativistic speed: \(\text{u=}\cfrac{\text{v+u}\prime }{1+\cfrac{v\text{u}\prime }{{c}^{2}}}\), where \(v\) is the relative velocity between two observers, \(u\) is the velocity of an object relative to one observer, and \(u\prime \) is the velocity relative to the other observer

relativistic Doppler effects

a change in wavelength of radiation that is moving relative to the observer; the wavelength of the radiation is longer (called a red shift) than that emitted by the source when the source moves away from the observer and shorter (called a blue shift) when the source moves toward the observer; the shifted wavelength is described by the equation

\({\lambda }_{\text{obs}}{\text{=λ}}_{s}\sqrt{\cfrac{1+\cfrac{u}{c}}{1-\cfrac{u}{c}}}\)

where \({\lambda }_{\text{obs}}\) is the observed wavelength, \({\lambda }_{s}\) is the source wavelength, and \(u\) is the velocity of the source to the observer