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
Angular velocity particle moves around the origin in a circle; but when there is no cross-radial component, it moves in a straight line from the origin. Since radial motion leaves the angle unchanged, only the cross-radial component of linear velocity contributes to angular velocity. The angular velocity "ω" is the rate of change of angle with respect to time, which can be computed from the cross-radial velocity as: Here the cross-radial speed formula_13 is the signed magnitude of formula_11, positive for counter-clockwise motion, negative for clockwise. Taking polar coordinates for the linear velocity v gives magnitude "v" (linear speed) and angle "θ"