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
Propellant mass fraction important role in the rocket equation: Where formula_10 is the ratio of final mass to initial mass (i.e., one minus the mass fraction), formula_11 is the change in the vehicle's velocity as a result of the fuel burn and formula_12 is the effective exhaust velocity (see below). The term effective exhaust velocity is defined as: where "I" is the fuel's specific impulse in seconds and "g" is the "standard acceleration of gravity" (note that this is not the local acceleration of gravity). To make a powered landing from orbit on a celestial body without an atmosphere requires the same mass