Variable-mass system events. The following derivation is for a body that is gaining mass (accretion). A body of time-varying mass "m" moves at a velocity v at an initial time "t". In the same instant, a particle of mass dm moves with velocity u. The initial momentum can be written as Now at a time "t" + d"t", let both the main body and the particle accrete into a body of velocity v + dv. Thus the new momentum of the system can be written as Since d"m"dv is the product of two small values, it can be ignored, meaning during d"t"
Product of group subsets "ST", then "ST" coincides with the semidirect product of "S" and "T". Finally, if both "S" and "T" are normal in "ST", then "ST" coincides with the direct product of "S" and "T". If "S" and "T" are subgroups whose intersection is the trivial subgroup (identity element) and additionally "ST" = "G", then "S" is called a complement of "T" and vice versa. By a (locally unambiguous) abuse of terminology, two subgroups that intersect only on the (otherwise obligatory) identity are sometimes called disjoint. A question that arises in the case of a non-trivial intersection between a normal subgroup "N"