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 velocity and the escape velocity. With a circular orbit, escape velocity is times the orbital speed. Thus the total escape velocity formula_29 when leaving one body orbiting a second and seeking to escape them both is, under simplified assumptions: where formula_31 for circular orbits. The last two columns will depend precisely where in orbit escape velocity is reached, as the orbits are not exactly circular (particularly Mercury and Pluto). Let "G" be the gravitational constant and let "M" be the mass of the earth (or other gravitating body) and "m" be the mass of the escaping body or projectile. At