section_path: Laws/Third law
section_name: Third law
paragraph_text: Third law
  
  Overly brief paraphrases of the third law, like "action equals reaction " might have caused confusion among generations of students: the "action" and "reaction" apply to different bodies. For example, consider a book at rest on a table. The Earth's gravity pulls down upon the book. The "reaction" to that "action" is not the support force from the table holding up the book, but the gravitational pull of the book acting on the Earth.
  
  Newton's third law relates to a more fundamental principle, the conservation of momentum. The latter remains true even in cases where Newton's statement does not, for instance when force fields as well as material bodies carry momentum, and when momentum is defined properly, in quantum mechanics as well. In Newtonian mechanics, if two bodies have momenta p 1 {\displaystyle \mathbf {p} _{1}} and p 2 {\displaystyle \mathbf {p} _{2}} respectively, then the total momentum of the pair is p = p 1 + p 2 {\displaystyle \mathbf {p} =\mathbf {p} _{1}+\mathbf {p} _{2}}, and the rate of change of p {\displaystyle \mathbf {p} } is d p d t = d p 1 d t + d p 2 d t. {\displaystyle {\frac {d\mathbf {p} }{dt}}={\frac {d\mathbf {p} _{1}}{dt}}+{\frac {d\mathbf {p} _{2}}{dt}}.} By Newton's second law, the first term is the total force upon the first body, and the second term is the total force upon the second body. If the two bodies are isolated from outside influences, the only force upon the first body can be that from the second, and vice versa. By Newton's third law, these forces have equal magnitude but opposite direction, so they cancel when added, and p {\displaystyle \mathbf {p} } is constant. Alternatively, if p {\displaystyle \mathbf {p} } is known to be constant, it follows that the forces have equal magnitude and opposite direction.