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
Sum and Product Puzzle "X" + "Y" and P knows the product "X" × "Y". Both S and P know all the information in this paragraph. The following conversation occurs: What are "X" and "Y"? The solution has "X" and "Y" as 4 and 13, with P initially knowing the product is 52 and S knowing the sum is 17. Initially P does not know the solution, since and S knows that P does not know the solution since all the possible sums to 17 within the constraints produce similarly ambiguous products. However, each can work out the solution by eliminating other possibilities following