Newton’s Third Law and Conservation of Linear Momentum

Newton’s third law and conservation of linear momentum are closely connected in Classical Mechanics, but they are not merely two different wordings of the same statement. Newton’s third law describes the forces exchanged by interacting bodies. Momentum conservation describes the evolution of an entire chosen system. The bridge between them is built using Newton’s second law and a careful separation of internal and external forces.

The PhysicsLibrary entry on Newton’s laws gives the third law in the familiar equal-and-opposite form,

while the momentum entry emphasizes that, when only internal forces are present, these paired forces cancel in the equation for the total momentum of the system. This article develops that connection explicitly and also explains its limits.

1. Momentum and Newton’s second law

For a particle of constant mass m, the linear momentum is

A particularly useful form of Newton’s second law is

This form is more fundamental for the present discussion than F = ma because conservation of momentum is a statement about the time derivative of momentum.

For two particles,

will denote the total linear momentum of the two-particle system. Differentiating gives

Newton’s second law can then be applied to each particle separately.

2. Newton’s third law as an interaction-pair statement

Suppose particle 1 and particle 2 interact. Let

denote the force exerted on particle 1 by particle 2, and let

denote the force exerted on particle 2 by particle 1.

For an ordinary Newtonian interaction satisfying the third-law form,

The two forces therefore have equal magnitudes and opposite directions, but they act on different bodies. They do not cancel in the Free-body diagram of either one particle. They cancel only after the two bodies are treated as one larger system.

This distinction prevents a common error. If particle 1 accelerates because particle 2 pushes it, one should not draw both F₁₂ and F₂₁ on particle 1. Only F₁₂ belongs on particle 1’s free-body diagram.

3. Two particles: the cancellation that produces momentum conservation

Let external forces F_1,ext and F_2,ext also act on the two particles. Newton’s second law gives

Add the equations:

Newton’s third law gives

Therefore

conceptually, where

This is the central result. Internal forces can transfer momentum from one particle to another, but in the Newtonian third-law model they cannot change the total momentum of the complete system.

4. Isolated system: conservation of total momentum

If the system is isolated so that

then

Hence

and therefore

For constant masses,

This familiar collision equation is therefore not an unrelated rule that must be memorized separately. Within Newtonian particle mechanics it follows from three steps:

1. Newton’s second law relates force to momentum change.
2. Newton’s third law makes internal interaction forces occur in equal-and-opposite pairs.
3. An isolated system has zero net external force.

Together these imply constant total momentum.

5. Momentum exchange: what the third law means dynamically

Suppose the particles interact for a time interval from t₀ to t₁ and external forces are negligible. Integrating Newton’s second law gives

Since

we obtain

Thus one particle’s momentum gain is exactly the other particle’s momentum loss:

This is the impulse-level version of the connection between Newton’s third law and momentum conservation.

6. Worked example: two skaters push apart

Two skaters initially rest on nearly frictionless ice. Let

After pushing apart, skater 1 moves at

Initially,

With negligible external horizontal force,

Thus

Solving,

The lighter skater moves faster. During the push the skaters exert equal-and-opposite forces on one another for the same interaction time, so their impulses are equal and opposite. The individual momenta change, but their vector sum remains zero.

7. Many particles

For a system of N particles, define

For particle i,

Summing over all particles gives

The internal sum contains paired terms

For Newtonian third-law interactions,

Thus all internal forces cancel pair by pair, leaving

8. Connection with the center of mass

For particles of total mass

the center of mass is

For constant masses,

Therefore

Combining this with the system momentum equation gives

Internal interactions can make the particles move violently relative to the center of mass, but they do not accelerate the center of mass of an isolated Newtonian system.

This is another expression of the same idea: internal forces redistribute momentum internally; external force controls the momentum of the system as a whole.

9. System boundaries matter

Momentum conservation is always a statement about a specified system. Consider a ball bouncing from a wall. The ball’s momentum changes, so the ball alone does not have conserved momentum during impact. The wall exerts an external force on the ball.

If the system is enlarged to include the ball, wall, and Earth, the contact forces between ball and wall become internal. The momentum transferred to Earth is usually too small to notice because Earth’s mass is enormous, but it is part of the total momentum balance.

This example illustrates a useful procedure:

1. choose the system;
2. identify which interactions are internal;
3. identify external forces;
4. apply dP∕dt = F_ext to that system.

Changing the system boundary can change which forces are called internal or external, but it does not change the physics.

10. Third law and conservation are closely related, but not identical

In elementary Newtonian mechanics it is natural to say that Newton’s third law leads to momentum conservation. That statement is correct for a system of particles whose internal forces occur as simultaneous equal-and-opposite pairs.

However, momentum conservation is the broader principle.

There are two important reasons to distinguish them.

10.1 Momentum conservation can be formulated without pairwise force language

In Lagrangian and hamiltonian mechanics, conservation of total momentum is associated with invariance under spatial translations. Noether’s theorem makes this relation systematic: continuous spatial-translation symmetry leads to conserved linear momentum.

Thus modern mechanics can derive momentum conservation from symmetry rather than taking pairwise third-law cancellation as the deepest explanation.

10.2 Fields can carry momentum

In Electromagnetism, momentum need not reside only in material particles. Electromagnetic fields themselves carry momentum. A simple instantaneous relation

between the mechanical forces on two charged bodies need not account for the entire momentum balance at every instant.

The correct conserved quantity is then schematically

Momentum can move temporarily into or out of the electromagnetic field while total momentum remains conserved.

This is why momentum conservation survives as a central principle even in theories where the elementary Newtonian third-law picture must be generalized.

11. Does momentum conservation imply Newton’s third law?

For an isolated two-particle Newtonian system,

Therefore

If Newton’s second law identifies these momentum rates with the mutual forces,

then

So, under these assumptions, conservation of the total momentum of an isolated two-particle system implies an equal-and-opposite force relation.

But the assumptions matter. If momentum can also be carried by fields or other degrees of freedom omitted from the two-particle description, conservation of the complete system does not require the two mechanical forces alone to be equal and opposite at every instant.

12. Strong and weak forms of the third law

The relation

is sometimes called the weak form of Newton’s third law: the forces are equal and opposite.

A stronger condition additionally requires the two forces to lie along the line joining the particles. Central Newtonian gravity is an example. The distinction becomes important for angular momentum: equal-and-opposite forces cancel total internal force, but conservation of orbital angular momentum from pair forces also requires the internal torques to cancel appropriately.

For linear momentum conservation, equal-and-opposite internal forces are the relevant ingredient.

13. Collisions and explosions

Momentum conservation is especially useful when internal forces are large but difficult to model. During a collision, contact forces can vary rapidly and may be unknown in detail. Nevertheless, if the collision time is short enough that external impulse is negligible,

The same reasoning applies to explosions and recoil. Internal forces may dramatically alter the momenta of individual pieces while leaving the vector sum unchanged.

This explains why momentum methods often solve collision problems more easily than direct force integration.

14. External impulse and the generalized momentum balance

If external forces are not negligible,

Integrating from t₀ to t₁ gives

Define the external impulse

Then

Momentum conservation is the special case

This formulation is often more useful experimentally than asking whether the external force is exactly zero at every instant.

15. Concept map

The logical relationship can be summarized as

The conceptual lesson is equally important:

Newton’s third law describes how momentum is exchanged internally between interacting parts of a Newtonian system. Conservation of momentum describes the balance of the complete system.

16. Connections to later physics

This bridge leads naturally to several deeper topics:

• center-of-mass motion: MA_CM = F_ext;
• collisions and impulse: conservation laws avoid detailed contact-force histories;
• angular momentum: internal torque cancellation provides the rotational analogue;
• Lagrangian mechanics and Noether’s theorem: spatial-translation symmetry explains conserved momentum;
• electromagnetism: fields possess momentum and complete the force-momentum balance;
• relativity and field theory: energy and momentum become components of a unified energy-momentum structure.

These topics generalize rather than discard the elementary Newtonian result.

References

References