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"
Angular velocity is formula_5, and the linear velocity is formula_6, so that formula_7. In the general case of a particle moving in the plane, the angular velocity is measured relative to a chosen center point, called the origin. The diagram shows the radius vector r from the origin "O" to the particle "P", with its polar coordinates formula_8. (All variables are functions of time "t".) The particle has linear velocity splitting as formula_9, with the radial component formula_10 parallel to the radius, and the cross-radial (or tangential or circular) component formula_11 perpendicular to the radius. When there is no radial component, the