Newton’s laws have engendered much discussion over several centuries. Today, the internet is awash with a plethora of information on this topic. We find many references to Newton’s laws, often discussions of various types of misunderstandings and ways to explain them. Here we present an intriguing example that shows an assumption hidden in Newton’s third law that is often overlooked. As is well known, the first law defines an inertial frame of reference and the second law determines the acceleration of a particle in such a frame due to an external force. The third law describes forces exerted on each other in a two-particle system, and allows us to extend the second law to a system of particles. Students are often taught that the three laws are independent. Here we present an example that challenges this assumption. At first glance, it seems to show that, at least for a special case, the third law follows from the second law. However, a careful examination of the assumptions demonstrates that is not quite the case. Ultimately, the example does illustrate the significance of the concept of mass in linking Newton’s dynamical principles.

1.
Interestingly, even though we call them Newton’s laws, Newton himself generously gave credit for the first and the second laws to Galileo. Please see p. 113 in Isaac Newton, The Principia: Mathematical Principles of Natural Philosophy, a new translation by I. Bernard Cohen and Anne Whitman, assisted by
Julia
Budenz
(
University of California Press
,
1999
), ISBN: 9780520088160.
2.
Newton’s Laws of Motion
,” in
W.
Thomson
(Lord Kelvin) and
P. G.
Tait
,
Treatise on Natural Philosophy
, Vol.
I
(
1867
), Sect. 242; and
Benjamin
Crowell
,
Newtonian Physics
(
2000
).
3.
Ernst
Mach
,
The Science Of Mechanics
, 4th ed. (
The Open Court Publishing Company
,
Chicago
,
1919
).
4.
This removes any need of introducing a normal force or weight in the force diagrams.
5.
Newton wrote, “The quantity of matter is the measure of the same, arising from its density and bulk conjointly.” Mach criticized this definition as being circular.
M.
Strauss
, the author of
Modern Physics and its Philosophy
(
Springer-Verlag
,
1972
), states on p.
125
that Newton’s above definition of mass could be dropped only if the additivity is introduced as an axiom. At least for homogeneous materials, the above definition involving densities implies the additivity of mass (to be able to compare densities).
6.
It works well for most forces encountered in an introductory physics course. It however fails for forces between two charged particles in relative motion unless electromagnetic fields are brought into the analysis. See
David J.
Griffiths
,
Introduction to Electrodynamics
, 4th ed. (
Pearson
2012
), Sect. 8.2.1.
7.
If v1 were strictly equal to v2, a “collision” would not happen. This collision is to be seen more as the limiting case of v1v2 = ε, with |ε| → 0.
8.
A very similar argument can be found in the following reference, where the author uses accelerations instead of velocities:
K. J.
McQueen
, “
Mass additivity and a priori entailment
,”
Synthese
192
,
1373
1392
(
2015
).
9.
N.
Feather
, “
Additivity of mass in Newtonian mechanics
,”
Am. J. Phys.
34
,
511
516
(
June
1966
).
AAPT members receive access to The Physics Teacher and the American Journal of Physics as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.