from one particle to the other; the fact that this depends on the frame shows how relative this is. With time reversed we have the situation of two objects pushed away from each other, e.g. shooting a projectile, or a rocket applying thrust (compare the derivation of the Tsiolkovsky rocket equation). Examples of collisions analyzed numerically Animal locomotion Collisions of an animal's foot or paw with the underlying substrate are generally termed ground reaction forces. These collisions are inelastic, as kinetic energy is not conserved. An important research topic in prosthetics is quantifying the forces generated during the foot-ground collisions associated with both disabled and non-disabled gait. This quantification typically requires subjects to walk across a force platform (sometimes called a "force plate") as well as detailed kinematic and dynamic (sometimes termed kinetic) analysis. Collisions used as an experimental tool Collisions can be used as an experimental technique to study material properties of objects and other physical phenomena. Space exploration An object may deliberately be made to crash-land on another celestial body, to do measurements and send them to Earth before being destroyed, or to allow instruments elsewhere to observe the effect. See e.g.: During Apollo 13, Apollo 14, Apollo 15, Apollo 16 and Apollo 17, the S-IVB (the rocket's third stage) was crashed into the Moon in order to perform seismic measurement used for characterizing the lunar core. Deep Impact SMART-1 - European Space Agency satellite Moon impact probe - ISRO probe and LCROSS with its spent Centaur Upper Stage - NASA Probe Double Asteroid Redirection Test for planetary defence Mathematical description of molecular collisions Let the linear, angular and internal momenta of a molecule be given by the set of r variables { pi }. The state of a molecule may then be described by the range δwi = δp1δp2δp3 ... δpr. There are many such ranges corresponding to different states; a specific state may be denoted by the index i. Two molecules undergoing a collision can thus be denoted by (i, j) (Such an ordered pair is sometimes known as a constellation.) It is convenient to suppose that two molecules exert a negligible effect on each other unless their center of gravity approach within a critical distance b. A collision therefore begins when the respective centers of gravity arrive at this critical distance, and is completed when they again reach this critical distance on their way apart.