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{excerpt}Every force exerted on one body by a second body is paired with another force of equal magnitude and opposite direction exerted on the second body by the first.{excerpt}

h4. Motivation for Concept

It is impossible for one object to alter the motion of another without in turn having its own motion altered.  The object that "instigated" the collision is at no fundamental advantage.  (Consider how much easier it would be to assign fault in a traffic accident without the Third Law.)  You can feel the consequences of Newton's Third Law by kicking a heavy object.  You can see the consequences by sitting next to another person on wheeled chairs and experimenting with what happens when one person pushes the other.

h4. Statement of the Law

h6. Newton's Statement

"To any action there is always an opposite and equal reaction; in other words, the actions of two bodies upon each other are always equal and always opposite in direction."  (_The Principia_ by I. Newton, translated by I.B. Cohen and A. Whitman.)

h6. Modern Statement

Through careful consideration, every [force] can be assigned two "participant" bodies.  One body is the originator of the force, the other is acted upon by the force.  Thus, any force can technically be assigned two indices.  A force *on* object A that *originates from* object B can be denoted _F_~AB~.  Newton's Third Law says that for any _F_~AB~, there will _of necessity_ be an _F_~BA~ related by:

{latex}\begin{large}\[ \vec{F}_{AB} = - \vec{F}_{BA}\]\end{large}{latex}

h4. Use in Problem Solving

h6. Free Body Diagrams

The most important consequence of Newton's 3rd Law is in setting up the equations of [Newton's 2nd Law|Newton's Second Law].  This is usually done with the aid of a [free body diagram].  If a problem requires you to construct free body diagrams for more than one object, it is vital to remember to apply Newton's 3rd Law to any forces acting on one body of interest that arise from another body of interest.  

{warning}It is important to remember (and to check) that when drawing the free body diagrams for multiple objects, forces only appear on the diagram of their target (not on the diagram of their originator).  Thus, _F_~AB~ should appear _only_ on object A's free body diagram, and _F_~BA~ should appear only on object B's free body diagram.{warning}


h6. Cancellation of Internal Forces

In the simultaneous equations resulting from [Newton's 2nd Law|Newton's Second Law] applied to more than one object, the forces that act on one body of interest that arise from another body of interest can _always_ be eliminated from the system trivially as a consequence of Newton's 3rd Law.  Thus, if the bodies are combined into a single [system], these [internal forces|internal force] drop out of the equations of [Newton's 2nd Law|Newton's Second Law] for that system.

h6. Conservation of Momentum

The constancy of the [momentum] of a system composed of many bodies that is isolated from [external influences|external force] is a direct consequence of Newton's 3rd Law applied to Newton's 2nd Law.  Because [internal forces|internal force] drop out of Newton's 2nd Law for the system, the system's momentum will be constant.

h4. The Horse and Cart "Paradox"

h6. Statement of the "Paradox"

Consider a trailer attached to a car.  If the car pulls on the trailer with a force _F_ in an attempt to drive away, Newton's 3rd Law guarantees that the trailer will pull back on the car with the same force _F_.  How, then, can the car pull the trailer?

h6. Resolution of the "Paradox"

The paradox is easily resolved if the forces are specified completely.  The car exerts a force _F_~TC~ *on the trailer* and the trailer exerts a force _F_~CT~ *on the car*.  The free body diagrams for the car and trailer including only these forces would look like:

!ThirdFBD1.png!

In one sense, we have already resolved the paradox.  The forces *do not* cancel, since they act on separate bodies!  In another sense, however, we still have to worry about the diagrams we have made.  It seems the car will be pulled backward while the trailer moves forward.  Indeed, in the absence of other forces, this would be the result (consider a tug-of-war in space).  We must not forget, however, that the engine is turning the car's wheels which rub against the ground.  Thus, the car's free body diagram must have (at least) the modification:

!ThirdFBD2.png!

where _F_~CG~ is the force acting on the car from the ground's friction with the car's tires.  We now see that as long as the friction between the car's tires and the ground (a result of the action of the car's engine trying to turn the wheels) is larger than _F_~CT~, both the car and the trailer will move forward.

It is important to remember (and to check) that when drawing the free body diagrams for multiple objects, forces only appear on the diagram of their target (not on the diagram of their originator). Thus, F_AB should appear only on object A's free body diagram, and F_BA should appear only on object B's free body diagram.