Proper velocity addition formula: where formula_3 is the beta factor given by formula_4. or in terms of logarithms: Proper velocity is useful for comparing the speed of objects with momentum per unit rest mass ("w") greater than lightspeed "c". The coordinate speed of such objects is generally near lightspeed, whereas proper velocity tells us how rapidly they are covering ground on "traveling-object clocks". This is important for example if, like some cosmic ray particles, the traveling objects have a finite lifetime. Proper velocity also clues us in to the object's momentum, which has no upper bound. For example, a 45 GeV electron
Velocity Moments image, formula_4 is the intensity of the pixel at the point formula_5 in the image, and formula_6 is the basis function. Cartesian velocity moments are based on these Cartesian moments. A Cartesian velocity moment formula_7 is defined by where formula_2 and formula_3 are again the dimensions of the image, formula_11 is the number of images in the sequence, and formula_12 is the intensity of the pixel at the point formula_5 in image formula_14. formula_15 is taken from Central moments, added so the equation is translation invariant, defined as where formula_17 is the formula_18 coordinate of the centre of mass for