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"
Brain-to-body mass ratio animals usually have larger brains than smaller animals); the relationship is not, however, linear. Small mammals such as mice may have a brain/body ratio similar to humans, while elephants have a comparatively lower brain/body ratio. In animals, it is thought that the larger the brain, the more brain weight will be available for more complex cognitive tasks. However, large animals need more neurons to represent their own bodies and control specific muscles; thus, relative rather than absolute brain size makes for a ranking of animals that better coincides with the observed complexity of animal behaviour. The relationship between brain-to-body mass