Newton's Third Law

Idea: Forces in nature always exist in pairs. Newton's third law states: For every action, there is an equal and opposite reaction. When two bodies interact:

$$\vec{F}_{2\,\,{\:\rm on}\,\,1}^{} \vec{F}_{1\,\,{\:\rm on}\,\,2}^{}$$ (4)

Where $\vec{F}_{\:\rm 2~on~1}^{}$ is the force exerted on body 1 by body 2 and $\vec{F}_{\:\rm 1~on~2}^{}$ is the force exerted on body 2 by body 1.

For Example: When an object falls towards the earth, the earth exerts a force on it that causes it to accelerate towards the earth. According to Newton's third law, the object exerts a force on the earth as well, and the earth accelerates towards the object. Why don't we feel the earth accelerate?
Solution:

$$\rightarrow \vec{F}_{\:\rm obj~on~earth}^{}$$ 2nd Law

$$\rightarrow \vec{F}_{\:\rm obj~on~earth}^{} \vec{F}_{\:\rm earth~on~obj}^{} \equiv \vec{w}$$ 3rd Law

$$\Rightarrow \vec{a}_{e}^{} \vec{w}$$

$$\Rightarrow \vec{a}_{e}^{} \left(\frac{m_{\:\rm obj}}{m_e}\right)g \ll$$

Conclusion: the acceleration of the earth is too small to detect because the mass of the earth is much larger than the mass of the object.