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"
Escape velocity to escape the gravitational influence, and that there will be no future sources of additional velocity (e.g., thrust), which would reduce the required instantaneous velocity. For a spherically symmetric, massive body such as a star, or planet, the escape velocity for that body, at a given distance, is calculated by the formula where "G" is the universal gravitational constant (), "M" the mass of the body to be escaped from, and "r" the distance from the center of mass of the body to the object. The relationship is independent of the mass of the object escaping the massive body. Conversely,