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
Escape velocity a body that falls under the force of gravitational attraction of mass "M", from infinity, starting with zero velocity, will strike the massive object with a velocity equal to its escape velocity given by the same formula. When given an initial speed formula_2 greater than the escape speed formula_3 the object will asymptotically approach the "hyperbolic excess speed" formula_4 satisfying the equation: In these equations atmospheric friction (air drag) is not taken into account. A rocket moving out of a gravity well does not actually need to attain escape velocity to escape, but could achieve the same result (escape) at