Newton's Third Law (Action and Reaction)

Every interaction between two systems produces forces equal in magnitude and opposite on different bodies.

Conceptual context

Newton's third law answers a question that looks simple but is easy to misread: when two bodies interact, what exact relation do the forces they exert on each other have? The answer is firm: they are always equal in magnitude and opposite in direction, and they act on different bodies. That last condition is the key to the whole topic.

This page organises the idea around two forces from the same interaction, the Force of A on B (the one A exerts on B) and the Force of B on A (the one B exerts on A), and around a particular case that appears in almost every contact problem: the pair of normal forces, formed by the Normal of A on B and the Normal of B on A. It also distinguishes a relation that is often confused: the Weight of a body and the Generic normal force that supports it. Both may have the same magnitude in equilibrium, but they do not form an action-reaction pair.

Understanding this law avoids the most widespread error of introductory dynamics: trying to cancel action and reaction as if they acted on the same body. They never do. For this reason, neither of them disappears from a correctly drawn free-body diagram.

🟢 Essential level

The essential intuition can be summed up in one sentence: forces never appear alone, always in pairs. If you push a wall, the wall pushes you back with the same intensity. If you walk, your feet push the ground backwards and the ground pushes you forwards. In both cases there are two forces, not one.

A classic example is that of two skaters on ice, A and B, at rest, who separate by pushing each other. The Force of A on B and the Force of B on A have exactly the same magnitude at every instant of the push. However, the skaters do not leave with the same speed: that depends on their masses, the Mass of skater A and the Mass of skater B, not on the forces of the pair.

This level also corrects a very common confusion: the normal force that supports a block and the weight of that same block seem to cancel on the diagram, and sometimes they even have the same magnitude, but they are not an action-reaction pair. The weight is the interaction between the block and the Earth; the normal force is the interaction between the block and the surface. They are two different pairs, and each has its own reaction on another body.

🔵 Formal level

Basic formulas

The general vector form states that the force B exerts on A is always the vector opposite to the one A exerts on B:

third law in vector form

This vector equality translates, axis by axis, into two scalar relations that make the work with free-body diagrams easier. On the x axis, the x component of force of B on A must be exactly the opposite of the x component of force of A on B:

third law along x

In the same way, on the y axis, the y component of force of B on A must be exactly the opposite of the y component of force of A on B:

Real student questions

If the forces are equal and opposite, why does the system move? Because they act on different bodies. Each body responds only to the force that acts on it, not to the one it exerts on the other.

Are weight and normal force an action-reaction pair? No. Both act on the same body. The third-law pair of the normal force is formed by the Normal of A on B and the Normal of B on A between the two surfaces in contact, and the pair of the weight links the body with the Earth.

Why do two skaters of different mass not leave with the same speed? Because the forces of the pair are equal, but each acceleration also depends on the mass of each skater: the acceleration of A is the force on A divided by its mass, and the same holds for B.

Final synthesis

Mastering the third law means distinguishing precisely who exerts the force and on whom it acts, never mixing an action-reaction pair with two forces that act on the same body, and remembering that equal forces do not produce equal effects when the masses involved are different.

When this distinction is secure, the dynamics of systems of several bodies becomes much more predictable and less prone to diagram errors.