The Principia's third law of motion: Original and generalized forms

In Newton's Principia, the third law of motion states: “to every action there is always opposed an equal reaction; or the mutual actions of two bodies upon each other are always equal, and directed to contrary parts.” The law only refers to the “bodies,” not other factors. The law does not take into account the inherent characteristics, nature, compositions, flexibility, rigidity, magnitude, distinctiveness of interacting bodies, etc. The bodies may be of steel, wood, rubber, cloth, wool, sponge, spring, etc. For all such bodies if the action is same, then the reaction must be the same. Furthermore, the comparative size and point of impact of target and projectile, etc., and roughness of surfaces play significant roles. In the qualitative explanation given after the definition to the law, Newton expressed action and reaction in terms of push or pull (force) and velocity. It can also be understood in terms of elastic collisions. It can be mathematically understood that in elastic collisions (coefficient of restitution, e = 1) the action and reaction are equal only under certain conditions. In one-dimensional elastic collisions, there are different cases, e.g., Action = −Reaction and Action = −1.002002 Reaction, etc. Thus, to take elusive factors into account, the third law of motion is generalized as: “To every action there is always an opposed reaction, which may or may not be equal in magnitude, depending upon the inherent characteristics of the process.” Replacing proportionality with a coefficient of proportionality K, the factors which are neglected in the original form of the law are accounted for. The value of K can be equal to, less than, or greater than unity. Thus action differs from the reaction. Some experiments are discussed which justify the generalised form of the law.