A Deductive Proof of Newton's Third Law
by Ken Amis

## 1. Background

Every physics student learns Newton's three laws of motion. It's tempting to consider that these are three separate and independent laws. That's not so. Both the first and third laws may be mathematically derived from the second law, as we will show.

The fact that the first law may be derived from the second has long been known. The second law, Fnet = ma, tells us that the net (vector sum) of all forces acting on a body is equal to the product of the body's mass and its vector acceleration. When the acceleration is zero, the net force must be zero. This is exactly the content of the first law.

## 2. The Third Law

Newton's third law is often considered "trivial", but it's more subtle than most students realize. It asserts that "If body A exerts a force on body B, then B exerts a force of equal size and opposite direction on A." It can be written: FAB = – FBA. The pair of forces in this law are often called an "action-reaction pair." Each force is said to be a "reaction" force of the other, though this language is mere window dressing. The terms "action" and "reaction" are often misleading to students and are best avoided in these discussions, for they aren't necessary.

Let's first consider the case of two bodies in contact. Each exerts a force on the other at the interface, or point of contact, where the bodies touch. If that point or interface is treated as a "body" of mass zero, then Newton's second law tells us that Fnet = 0a, so Fnet = 0. So the net force on a body of zero mass is always zero, whatever forces act upon it. Therefore if only two forces act on a body of mass zero, they must add to zero, and therefore must be equal size and oppositely directed. This establishes Newton's third law.

Restating this may make it clearer. Any force can be decomposed into two parts. In this case the net force on the interface may be considered the sum of: (1) The net force due to A acting on the interface, and (2) the net force due to B acting on the interface. Two bodies in contact are equivalent to two bodies with a zero mass body between them at the point of contact. We have shown that these two forces add to zero, so they must be forces of equal size and opposite direction. Q.E.D.

## 3. A Closer look

If that seems too "pat" for your tastes, we can make the argument more rigorous. Consider three balls contacting each other as shown in Fig. 1A. We show only the contact forces of the two larger balls acting on the smaller one. Of course the smaller one exerts equal and oppositely directed forces on the larger ones as well. Now consider the limiting case as the small ball is made smaller, as in Fig. 1B, and finally in Fig. 1C the small one has shrunk to zero dimension—a point. The initially unequal size forces shown have necessarily become equal. They are now also colinear and oppositely directed.

 Fig. 1. Three bodies in contact. The forces exerted on the smaller middle one are initially unequal in size or direction. In the limit as the middle body's size and mass both go to zero, those two forces become equal, opposite and colinear at the point of contact. Though Fig. 1 shows the case of compression at the point of contact, the argument applies equally well to forces in the opposite direction, for example, gravitational attraction.

## 4. Surface contact

When the bodies contact along a surface, we can subdivide the surface into infinitesimal pieces that may be treated as points. The argument of section 2 may then be applied, concluding that the force of A acting on B is of equal size and opposite direction to the force of B acting on A, and these forces are coliniear, so they produce no torque. Now integrating over the whole surface of contact we find that the net force of A acting on B is also of equal size and opposite size to the net force of B acting on A, and the net torque due to all forces is zero, which means that FAB = – FBA. Again, we have established Newton's third law.

For a concrete illustration, consider two bodies in contact. Now place a piece of paper separating them at the point of contact. The fact that the paper has much smaller mass than the two bodies ensures that the net force on the paper is very small, and the forces the two bodies exert on it are nearly equal and opposite. This example may be useful in teaching this concept to students.

## 5. Final Generalization

So far we have considered only bodies in contact. What about forces that act at a distance, such as gravitational, electric and magnetic forces? Here's where our approach to this problem allows really profound insights.

 Fig. 2. Schematic illustration of two separated bodies A and B interacting with space in accordance with Newton's third law. Forces on these bodies are shown. If there's space between two bodies, of whatever extent, but zero mass, then treating space as "the third body in the middle" yields the same result as above! You didn't expect it to be that simple, did you?

## 6. New Insights From This Approach

Consider the implications flowing from this new approach. If Newton's third law is universally true, it is telling us that the space between objects must indeed have zero mass. Remember all those years physicists wasted on the idea of a substance called the "luminiferous ether" that "fills all of space".  If they'd only had the benefit of the proof we've outlined above they'd have realized that this ether must have exactly zero mass. Then, if they really believed Newton's third law, they wouldn't have bothered with the (now abandoned) notion of the ether. They'd have realized that their ether was experimentally indistinguishable from nothing. 

Though the luminiferous ether idea has disappeared from textbooks, seldom rating even a footnote, modern physics has introduced subtler and sneakier ways to give structure and substance to space. These have fancy names like "vacuum states". If any of this new stuff supposedly "in" space has mass, or if space itself has mass, then careful measurements of forces between interacting bodies should reveal that fact. Any inequality of action and reaction forces on bodies interacting through intervening space would reveal the mass of space.

Critics of this interpretation of Newton's Third Law may object to treating space as a "massless body". Why should this be so alarming? Physicists have entertained even crazier concepts and even incorporated them into their theories. In the 20th century physicists quite comfortably lived with the notion of massless neutrinos.

This analysis does not appear in Newton's writings, yet he invented the calculus, and surely had some grasp of limiting processes.  Why did he split his revolutionary idea into three distinct parts? Could it be that he didn't realize that the three laws were really one? Could he have held back this important insight so that competitors couldn't easily follow his "giant's foosteps"?

## 8. Conclusion

It's about time we quit speaking of "Newton's three laws" and simply refer to this important idea as "Newton's law of mechanics." That's two fewer laws students will need to cram for exams. It's often said that you can pass an elementary course in physics if only you know Newton's laws of mechanics and all of their logical consequences. Those consequences include the conservation laws of energy and momentum. There may be something to that.

• Ken Amis

## Endnotes

1. Swenson, Loyd S. The Ethereal Aether, a History of the Michelson-Morley-Miller Aether-Drift Experiments. University of Texas Press, 1972.

2. Newton, Sir Isaac. The Mathematical Principles of Natural Philosophy. 1729.

3. Physicists now play with the notions of dark matter and dark energy supposedly filling all of space, accounting for subtle data about the motions of distant stars and galaxies. This in no way invalidates the arguments given here. It just makes the calculations more difficult.

© 2002 by Ken Amis and Donald E. Simanek. Permission for reproduction and use of this entire document is granted for educational non-profit purposes only.