# Physics with Calculus/Mechanics/Newton's Third Law

Newton's third law states: For every force, there is an equal and opposite reaction force (${\vec {F}}_{\mathrm {a} b}=-{\vec {F}}_{\mathrm {b} a}$ ). This is the basis of "conservation of momentum" to be covered later.

## Third Law (Equal and Opposite Reactions)

Now that we have the concept of force, we can begin exploring how it effects the world around us. Consider this. You are most probably sitting in a chair. How many forces are acting on you at this moment? Since you are not accelerating in any direction it would be easy to think that there are no forces, but that would be incorrect. Instead there are a great many forces that are all balanced. For instance, there is the force of gravity pulling you down. If you were magically suspended there above the Earth, the force of gravity would pull you swiftly, and ungraciously, to the ground. But you do not fall to the ground, instead you exert a force on the seat of the chair.

Here is where Newton's Third Law comes into play. You are exerting a force on the chair equal to your weight (let us assume for a moment that all your weight is resting on the chair). This is equal to:

${\vec {F}}=m{\vec {g}}\$

This is the force that you apply to the seat of your chair. However, thanks to Newton's Third Law, this is also the force that the chair applies upward on you. Because of this, the force of gravity pulling you down is countered by the force of the chair pushing you up, and you do not go sprawling all over the floor.

This is a hard concept, because it always applies, even when you might think it does not. Let us deal with a human attempting to catch a freight train. When the freight train hits the human, which has more force applied to it? The answer is, from the title of this section, that the forces are equal. The force applied to the human by the train is the same as the force applied to the train by the human. Now, you should note that a few thousand Newtons of force has a much greater effect on a human than it does on a barreling freight train, but the force is the same. It is the objects that are different, a trick that is often used to fool students on examinations.

Another example is this. You are standing on a sheet of frictionless ice. You can thrash around, you can flail your arms, but you cannot move your position (your center of mass to be precise). It is because if you try to move one way, and you apply a force to yourself, you are are simultaneously pushing yourself forwards and backwards, which results in nothing. Equivalently, if you put yourself in a basket, and pull on the handles, you can't lift yourself because while your are pulling up with your hands, you are pushing down with your feet. If you shoot a gun, you feel recoil. All of these are examples of the third law.

The third law, however, is not really a law because it is not true! In electrodynamics, it turns out, things can lift themselves by their bootstraps. However, the conservation of momentum, which is very similar to the third law, remains true.