Proper velocity by Hermann Minkowski's flat-space metric equation . Here a reference map frame of yardsticks and synchronized clocks define map position x and map time "t" respectively, and the "d" preceding a coordinate means infinitesimal change. A bit of manipulation allows one to show that proper velocity where as usual coordinate velocity . Thus finite "w" ensures that "v" is less than lightspeed "c". By grouping "γ" with v in the expression for relativistic momentum p, proper velocity also extends the Newtonian form of momentum as mass times velocity to high speeds without a need for relativistic mass. The proper velocity
Sector mass spectrometer is simple. The physics are described by a single equation called the Lorentz force law. This equation is the fundamental equation of all mass spectrometric techniques and applies in non-linear, non-homogeneous cases too and is an important equation in the field of electrodynamics in general. where E is the electric field strength, B is the magnetic field induction, "q" is the charge of the particle, v is its current velocity (expressed as a vector), and × is the cross product. So the force on an ion in a linear homogeous electric field (an electric sector) is: in the direction of