Newton's Third Law

One commonly finds Newton's third law stated as:

The Third Law:

For every force (action) there is an equal and opposite force (reaction).
₁₂ = - ₂₁

Here ₁₂is the force exerted by object 1 on object 2, and ₂₁ is the force exerted by object 2 on object 1. The third law concerns two forces that act at the same time. The third law is often used improperly to imply that anything that is done in society or economics will have a reaction in the other direction at some later time.

Which force is the action and which is the reaction is arbitrary. At first glance the third law may seem to contradict the second law; so one must remember that:

The third law applies to two bodies.
The second law applies to one body.

What we mean by this is further illustrated in this little Gedankenexperiment, a thought experiment:

Example:

Consider a free-falling apple.

The Earth exerts a force on the apple.

The apple exerts a force on the Earth.

The forces are equal and opposite.

BUT: The accelerations are obviously different! The Earth does not seem to move at all, whereas the apple obviously falls down towards the Earth. How does this reconcile with the Third law stated above?
Answer:
The acceleration of an object is
\[ \rm \mathbf{a = \frac{F}{m}} \]
(from Newton's Second Law). And since the mass of the Earth is MUCH bigger, its acceleration is MUCH smaller, even though (or perhaps exactly because) the magnitude of the force acting on both objects is exactly the same.