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
Mass ratio from Tsiolkovsky's rocket equation: where This equation can be rewritten in the following equivalent form: The fraction on the left-hand side of this equation is the rocket's mass ratio by definition. This equation indicates that a Δv of formula_4 times the exhaust velocity requires a mass ratio of formula_5. For instance, for a vehicle to achieve a formula_6 of 2.5 times its exhaust velocity would require a mass ratio of formula_7 (approximately 12.2). One could say that a "velocity ratio" of formula_4 requires a mass ratio of formula_5. Sutton defines the mass ratio inversely as: In this case, the values