The  english physicist, mathematician and astronomer Sir Isaac Newton (1642-1727), basen on the studies of Galileo and Descartes, published in 1684 the first great work in Physics: Philosophiæ Naturalis Principia Mathematica, also known as Principia. In the first of the three parts in which the work is divided, he exposes in three laws of motion the relations between forces and their dynamic effects: the laws of motion.

Newton´s third Law or action and reaction principle establishes that for every action force, there is an equal and opposite reaction force. We will go in deph through the following three points:

Shall we start?

Concept

Imagine several marbles, all of them with the same mass. When you throw a marble against another, probably the first one will stop and the second one will get a very similar velocity than the first marble had.

Marbles and action - reaction principle.

On the left, the blue marble goes at velocity $\stackrel{\to }{v}$. On the right, the blue marble remains still after hitting the red marble, of equal mass than the first one. The red, then, is set in motion at a very similar velocity to that the blue one had, $\stackrel{\to }{v}$.

From this simple example, you can verify that, for both marbles to modify their velocity have had to go through to forces. Since we can suppose marbles are isolated (they don´t interact with any other element), forces have only appeared during the blow. It seems a force over the hit marble has appeared during this action, which makes it set in motion. Therefore, it seems clearly that, since the "hitting" marble stops, it must experiment a reaction with a similar force but opposite direction.

We can now give a definition for Newton´s third law.

Definition

When a body A exert a force on another body B. B will react exerting another force on A with equal magnitude and opposite direction. The first of the forces recieves the name of action force  and the second reaction force where:

$\stackrel{\to }{{F}_{AB}}=-\stackrel{\to }{{F}_{BA}}\phantom{\rule{0ex}{0ex}}{F}_{AB}={F}_{BA}$

• $\stackrel{\to }{{F}_{AB}}$: Its the action force of A on B and it´s unit of measure is in the S.I. is the newton (N).
• $\stackrel{\to }{{F}_{BA}}$: It´s the reaction force of B on A and its unit of measure in the S.I. is also the newton (N).

Some important observations are that:

• Action and reaction forces have an equal and opposite direction so... why don´t they cancel each other?
• These forces don´t cancel each other because they are applied on different bodies.

Action and reaction forces.

When you push a box, the force which is applied acts on the box (in blue), This force is the responsible for the box to be moved. At the same time , the box exerts a reaction force on yourself (in red) which is the responsible for you to feel, on your hand, a resistance against the box´s movement.

• An exception to the previous point is the case we find when we study the two bodies as one (the particles making a  rigid body, for example). In this situation they would cancel each other, indeed.
• The principle is applicable not only to contact interactions, but to distance forces also. For instance, the Sun, due to its mass, exert an attraction force on Earth, but that also applies an attraction force on the Sun of equal (in size) and opposite direction. Then... why does the Earth orbit around the Sun and not the other way around? The truth is that actually, both orbit around a common point: the center of mass of the two of them. Because the mass of the Sun is bigger than the Earth´s mass, this point in inside the Sun, and the only perceptible orbit is our orbit turning around the Sun.
• The previous example also emphasizes that equal forces don´t imply equal effects. So, the effect of the force on Earth is much clearer than on the Sun.
• The principle assumes that forces occur simultaneously and are spread instantly. This is not easy to refut in the contact interactions. In the case of electromagnetic interaction, or even gravitatory, Einstein´s general relativity theory marks the maximum speed the interactions can be transmitted. This opened up a new horizon in the study of the dynamic that took to redifine some of the concepts we have introduced... but this is a long story that, for the moment, is far from the range of this level.

couple consists of two parallel forces that are equal in magnitude, opposite in sense and do not share a line of action.

Applications

Your day to day is full of examples where you use the action-reaction principle to get on well in your environment. In the subject Newton´s law applications we will study many of these examples, we start here with some of them, though:

Standing up

When you are standing up, the Earth applies an attraction over you, but... why don´t you sink? Due to the reaction of the ground. It has equal value and opposite direction. As it is reflected on the following figure, we can distinguish:

• : The attraction force exerted by the Earth and acts on you. We call it weight and it´s an action-at-a-distance force.
• : As a reaction, you also attract the Earth. This force is also a distance force acting on Earth. Of course, the Earth will barely notice this force but it exists...
• : It is the force you apply by contact on ground, just for being standing up. Its value and direction is equal to ${\stackrel{\to }{F}}_{\text{Earth-you}}$, but it acts on the ground.
• : It is the reaction to the previous force. It acts on you and it´s call normal force. Observe the normal force is not the reaction to the weight-force but the force you exert on ground and acts on ground. Bear in mind that action-reaction forces acts on different bodies. Otherwise, motion would not be possible.

Diagram of action and reaction forces

TODOThe figure shows the forces exerted on the arth and on you when standing up. The first couple action-reaction is formedLa figura muestra las fuerzas que aparecen sobre la Tierra y sobre ti cuando permanece de pié. El primer par acción-reacción está formado por F-Earth-you and F you-Earth. They are action at a distance forces. On the other hand, there is the couple action-reaction formed by F ground-you and F you-ground. They are forces of action by contact.TODO

Walk

When you walk, and thanks to the frictional force, you "propel the Earth backwards". The reaction of the Earth on your feet is to propel them forward.

Action- reaction when walking

Even though our detective is looking for a clue everywhere, he has the answer closer than he thinks. It is the push he exerts on the ground the one that causes a force on it, in red, whose reaction on the detective, in blue, is the responsible of his shift.

Run

In the starting line the athletes use the track to propel themselves. They do it by exerting a force against the track, so the reaction of it gives them the drive desired.

Action - reaction in athletics

In the start line, athletes uses the start block to launch themselves and counter their own weight, assisting the advance force in the most horizontal way possible, being the good relation between the weight and muscular performance essential for the athletic training.

Swim

On the same way, when you swim you move the water backwards, because of this, the water pushes you forward. When you get to the end of the pool and you want to turn around, you will probably propel yourself hard with your feet against the wall. The reaction of the wall on your feet is what let you to "gather momentum".

Action - reaction in swimming

When you swim, your hands, and also your feet, exert a force on the water, in red, whose reaction, in green, propels you forward.  The rest of your body must be set in the way it opposes the least water resistance possible, for this advance is as fastest and comfortable as possible.

To kick a ball

When you play football and kick the ball, it´s the force on the ball the one that makes the ball to be shot off. This, at the same time, exert a reaction force on your foot that it will make the foot to go backwads from the impact spot.

Action - reaction in football

The action force, in blue, and reaction, in green, make the ball to be shot off possible. What the "king of sports" would be without those forces?