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
Variable-mass system on a body (F = 0) the variable-mass system motion equation reduces to If the velocity of the ejected propellant, v, is assumed have the opposite direction as the rocket's acceleration, dv/d"t", the scalar equivalent of this equation can be written as from which d"t" can be canceled out to give Integration by separation of variables gives By rearranging and letting Δ"v" = "v" - "v", one arrives at the standard form of the ideal rocket equation: where "m" is the initial total mass, including propellant, "m" is the final total mass, "v" is the effective exhaust velocity (often denoted