Confusion Regarding Newton’s Third Law of Motion

Recently on the Modeling Instruction listserv there was a discussion about difficulties that students have with Newton’s Third Law and how teachers might best address these difficulties. It appeared to me (and I contributed to the discussion saying so) that some of the language being used by teachers was leading to  student confusion. It also appears to me that previous student experience with how the the Third Law is commonly explained (even well before high school) might be responsible. Finally, there are several different difficulties that students might be experiencing (arising from different conceptual issues) when they fail to use Newton’s Third Law correctly.

In this post I will address how Newton’s Third Law is commonly taught in high school (as well as introductory college courses), what the Third Law actually says, what it means and how all of this is often not at all clear to  students (or even teachers!). I’ll save for a separate blog post discussion of ideas on how to address student difficulties with the Third Law. In case you don’t have time to read all of this post, here is the short version:

1. Newton’s Third Law is taught in first year courses as The Principle of Reciprocity, not as a law of motion. While this is a possible source of confusion, I believe it is still a good idea.
2. The popular version of the Third Law involving the words “action” and “reaction” is actually a law of motion. The words action and reaction as used by Newton do not refer to forces. The use of the phrases “action force” and “reaction force” is a definite source of confusion.
3. Much of the easily available information on Newton’s Third Law is presented incorrectly, and serves to reinforce student misconceptions or even confuse students further.
4. Students have heard the Third Law presented as the Principle of Reciprocity disguised as a law of motion by teachers who don’t understand the difference, yet students think this is the one law they really do understand (since they can recite it) and are thus loathe to give it up!

How We Teach the Third Law

In first year courses, Newton’s Third Law is usually taught as a relationship between the forces two objects exert on one another. These forces are a result of a single interaction between the two objects (for instance, a gravitational interaction). One precise description of this particular manifestation of the Third Law is something like:

If object A exerts a force on object B, then object B exerts the same kind of force on object A, with the same magnitude and opposite direction.

As stated above, this Law says nothing about the motion (or changes in motion) of either object, yet it is the third of what are often called “Newton’s Laws of Motion.” Already we start to see why students get confused. Often students try to use the phrase “same magnitude and opposite direction” to attempt to come to some conclusion about the motion of a single object, leading them to falsely assume both forces in question are exerted upon that single object.

The concept stated above is more accurately described as “The Reciprocity Principle,” in that it describes a relationship between the two forces involved in a single interaction.  The Reciprocity Principle, as so stated, is not even universally true. Take, for instance, two protons, proton A moving in the positive x-direction, proton B moving in the positive y-direction. Calculate the instantaneous electric and magnetic forces that proton A exerts on proton B. Now calculate the instantaneous electric and magnetic forces that proton B exerts on proton A. Do this and you’ll see the point–there exists no reciprocity for those magnetic forces between individual charged particles!

So was Newton wrong? I’ll comment on that in the next section. For now let’s concentrate on why we would phrase Newton’s Third Law of Motion in a way that isn’t a law of motion, and doesn’t actually hold true for magnetic forces. The point of introducing Newton’s Third Law this way is that the Law of Reciprocity is a very good model for almost all interactions we encounter during a first year of physics at the high school or college level. The Principle of Reciprocity gives our students a tool with which to attack more complex situations involving several objects interacting within a system. As a bonus, we give ourselves a nice way to introduce the Principle of the Conservation of Momentum.These are good pedagogical reasons to teach the Principle of Reciprocity, rather than the Third Law as stated by Newton.

What Did Newton Actually Write?

Newton’s Third Law as found in Principia Mathematica is:

Lex III: Actioni contrariam semper et æqualem esse reactionem: sive corporum duorum actiones in se mutuo semper esse æquales et in partes contrarias dirigi.

Great, it’s in Latin. Not to fear, Wikipedia has the following translation (Wikipedia article on Newton’s Laws of Motion) which is not attributed:

To every action there is always an equal and opposite reaction: or the forces of two bodies on each other are always equal and are directed in opposite directions.

This is somewhat the version that is often quoted, leading to explanations involving “action forces” and “reaction forces.” After the confusing stuff about action and reaction (just what does Newton mean by these words? what do WE mean by those words?), it says something about forces. But wait. Just below this translation in the same article, a different, yet this time attributed, translation is given:

LAW III: To every action there is always opposed an equal reaction: or the mutual actions of two bodies upon each other are always equal, and directed to contrary parts.

What are we to make of this? The second translation of the same text does not include the word “forces,” but rather uses the word “actions.” I consulted a Latin scholar far more erudite than myself: my son. He pointed out that the word Newton uses for force (vis) is not at all present in the above statement. Thus the second translation seems to more accurately reflect what Newton wrote. Still, what does Newton mean by “action?” Let’s look to the paragraph following this sentence (again, the translation is taken from the same Wikipedia article and the same attributed source):

Whatever draws or presses another is as much drawn or pressed by that other. If you press a stone with your finger, the finger is also pressed by the stone. If a horse draws a stone tied to a rope, the horse (if I may so say) will be equally drawn back towards the stone: for the distended rope, by the same endeavour to relax or unbend itself, will draw the horse as much towards the stone, as it does the stone towards the horse, and will obstruct the progress of the one as much as it advances that of the other. If a body impinges upon another, and by its force changes the motion of the other, that body also (because of the equality of the mutual pressure) will undergo an equal change, in its own motion, toward the contrary part. The changes made by these actions are equal, not in the velocities but in the motions of the bodies; that is to say, if the bodies are not hindered by any other impediments. For, as the motions are equally changed, the changes of the velocities made toward contrary parts are reciprocally proportional to the bodies. This law takes place also in attractions, as will be proved in the next scholium.

Gracious me. Newton’s explanation starts out talking about forces and pressures, speaks of obstructing and advancing “progress,” and finally ends up talking about what appears to be momentum. I’ll say right here that I am unimpressed with the clarity of Newton’s explanation. I go back and forth between thinking action/reaction should be read as the change in motion as opposed to being read as the thing that changes the motion. But rather than pick nits (and criticize an author who can’t defend himself), let’s zero in on what Newton seems most intent upon telling us in this, his third law of motion: there is a specific relationship between the changes in the motions of two interacting objects. Hence, this is a law of motion.

In fact, it appears that Newton’s Third Law is a statement of the conservation of momentum for the situation of two isolated (“if the bodies are not hindered by any other impediments”), interacting objects. Although Newton didn’t know about field momentum, the momentum interpretation of the Third Law almost works for the case of two interacting protons that I cited above. If you take proton A, proton B and the electromagnetic field as your system, the momentum of the system is indeed constant, but this is no longer a two-body problem. The Principle of Reciprocity, by focusing on forces rather than momentum, misses the momentum that is gained by the electromagnetic field.

Besides noting that Newton’s version of the Third Law being an actual law of motion, I’d like to make the case that the terms “action” and “reaction” should be dropped because of the confusion they cause. Regardless of what Newton meant, today these words have no meaning in terms of forces or changes in momentum. Some of the discussion on the Modeling Instruction listserv revolved around students mistaking the “reaction” for the change in motion. Well this makes perfect sense, given our current understanding of the word reaction! We could clearly define the words action and reaction for use in the physics community (much as we clearly define work, energy, momentum, etc.), but we rarely use these words except to talk about the Third Law. Introducing new terminology only to abandon it soon afterwards is simply confusing to the student. Don’t do it. The only good use of action/reaction is for the name of a blog.

What Can Students Find on the Internet About the Third Law?

Sadly, much of the help for Newton’s Third Law that is available for students on the internet is poorly explained or just plain wrong. The confused student is very likely to seek help where help is supposed to be. Why wouldn’t a student struggling with Newton’s Third Law not simply pop “newton’s laws of motion” into a search bar? If they do, they will find the world stacked against them.

The first hit on just such a Google search results in a college site (must be vetted, no?) that gives the standard unhelpful  “For every action there is an equal and opposite reaction” followed by a single example that talks about motion (not forces, not changes in motion, but motion). This example does nothing to help explain the Third Law. So the student keeps looking. “Hey, next is rice.edu? Mr. Hammond went to Rice, this should be good.” The same unhelpful phrase is given followed by an example of a rocket. However the example misidentifies the appropriate force-pair and seems to indicate a rocket cannot accelerate unless it has the ground to push off of… confusing. Go on to the third hit, which leads to Wikipedia. Lots of symbols, calculus right off the bat, Latin, blah, blah, blah, nothing helpful to the beginning student. The fourth hit: NASA. Now we’re making progress! But the worksheets from NASA all include the misconception that both forces described by the Third Law are exerted on the same object! That’s right, the NASA site is dead wrong! Fifth site: Discovery Channel! Yay, TV! Yet the physics is all wrong, making the same mistake as the NASA site. Sixth site: physics4kids.com, a terrible website full of incorrect physics, including an incorrect explanation of the Third Law (same mistake as the NASA site made). Finally, on the seventh and eighth hits, we get some explanations that are correct and might be useful to the beginning student. How many students are going to get past the unhelpful and incorrect explanations (which probably coincide with their current misconceptions)?

Oh, you can go look at Khan Academy. At least they changed their wrong explanation after some physics teacher complained (who was that?). But for reasons I’ll explain in my follow up post to this one, passive explanations don’t really get you too far unless you are already almost there.

What Do Kids Bring to the Classroom Regarding the Third Law?

Unfortunately, the frequency with which I hear students spout “for every action there is an equal and opposite reaction!” whenever they see any two forces pointing in opposite directions with equal magnitudes tells me that this kindergarten version of the Third Law is omnipresent. The students are presented with this version in middle school or on TV science programs, and they have probably had at least one confused adult give an incorrect explanation of what it means. For such a widely memorized tidbit, there sure is a lot of misunderstanding about what it means! To make things worse, many kids come into physics thinking that this is the ONLY bit of physics they already understand!

In many school settings, the Third Law is stated simply as “for every action there is an equal and opposite reaction.” This might be fine if the explanation that followed developed action and reaction as changes in momentum. But nearly universally, this version of Newton’s Third Law is followed by a lot of talk about “action forces” and “reaction forces” and very little talk about what objects are exerting forces and what objects are being subjected to forces. Most middle school textbook explanations I’ve seen are simply impossible to understand.

Thus students tend to make the following mistakes regarding the Third Law:

1. They believe that the force a larger object exerts on a smaller object is larger than the force the smaller object exerts on the larger object. A big truck must exert greater force than a small car. A long rope must exert a greater force than a small rope. (Force-pair relationships are effected by size.)
2. They believe that the force exerted by a fast object, when colliding with a slow (or stationary) object, must exert a greater force than the slower  object exerts on the faster object. (Force-pair relationships are effected by relative speed.)
3. They buy into the Third Law only for objects at constant speed. (Force-pair relationships are effected by absolute speed.)
4. They think that the law refers to forces exerted upon (and the motion of) a single object. This mistake is understandable given the massive amounts of misinformation to which the students have been subjected. (Force-pair misidentified/not identified.)
5. They think that the Third Law no longer applies when one of the interacting objects breaks. (Force-pair relationships are effected by the strength of the materials involved.)
6. They can properly use the Third Law, but don’t believe that it works in the “real world.” (Physics world and real world do not overlap. Getting an A means saying some stuff just for the teacher’s benefit: “Those forces are equal and, yes, Beloved is my favorite novel.”)
7. They can state and use the Third Law, but fail to see when it might be helpful in various contexts. (Poor transfer of knowledge.)

I have some ideas about how to address student misunderstandings of the Third Law which I’ll share in a follow-up post. Some of the students’ misconceptions are actually astute observations hindered by lack of a consistent conceptual framework for incorporating those observations. That is, rather than just being ignorant, the students are trying quite hard (and almost successfully!) to make sense of the world. This should be a very strong position from which to start!