2.7 Newton’s Third Law

Critical Questions:

  • If two astronauts are floating in space and one of them shoves the other, how come they both float away in opposite directions?
  • What makes a car move forwards?

Walking is such a simple task that most mammals can do it mere hours after being born. (It takes humans a few years to figure it out, but we get there eventually.)

baby elephant
Yeeeaaah I’m walkin’

It’s so simple that the average person, unless they are a very special type of person, does not think very hard about the physics of walking.

But think about it: how does it really work? One leg moves forward, the other moves back, and somehow your entire body gets propelled in the direction you want to travel. It’s a great mystery!

Or perhaps you’re rolling your eyes right now. “Come on,” you’re saying, “don’t be an imbecile! Each time you move a leg backwards, you push yourself forwards at the same time. Easy!”

My first response is that you should calm down. My second response is that you’re right, of course. But let’s take what we know about physics so far and apply it to walking. After all, we know exactly how motion works in the absence of forces (Newton’s First Law), and we know what effects forces have on an object’s motion (Newton’s Second Law). So explaining how walking works should indeed be simple.

Let’s consider the case of a person standing still and then beginning to walk. You can try it yourself to get a feel for exactly what your leg is doing. (I’m being serious about that.) Because this is a case of forwards acceleration (your velocity changed from zero to about 5 km/h), we know that there must be some force pushing you forwards. You’ve already told me that this force is provided by your leg. But if you’ll think back to a previous section, you’ll remember that an object will only move forwards due to an external force. (We discussed the example of pushing on your own back, and we said that it has no effect because your arm comes with you when you move, and so its push is actually an internal force.)

In other words, some object that is not a part of you must be pushing you forwards – this is the only possible cause of your forwards acceleration. But what could that something be?

To answer this question, we should talk a bit about forces again. Classical Mechanics defines two kinds of forces: contact forces and non-contact forces. A contact force occurs when two objects are in contact with each other, such as when a hand pushes a door open. The four fundamental forces, on the other hand, are non-contact forces, because they act at a distance: for example, the moon is gravitationally attracted to the Earth even though it is far up in space.[1. Of course, we have already discussed the fact that even contact forces are really just fundamental forces in disguise. In macro-sized situations, however, it is perfectly valid to pretend that contact is actually occurring and to ignore the huge numbers of particles interacting with each other.]

Now we can return to your walking problem and try once again to discover what force is causing you to accelerate. We can first consider the non-contact forces: gravity is keeping you on the surface of the Earth, but not pushing you in any other direction; there are probably no large magnets in front of you; and the other two fundamental forces act only on subatomic particles and don’t have a long enough range to affect this situation.

So we’re left with some kind of contact force. But the only two things you are in contact with while walking are the ground and the air around you. You don’t use the air to move… And so, by process of elimination, we know that the ground must be pushing you forwards.

This should seem like a surprising thing to say, if you consider it carefully. After all, we are very much used to the idea that forces only occur when something or someone does something – when a person pushes something, or when an engine runs, or whatever. When a person walks along the ground, that ground seems remarkably uninterested in the person on top of it. So why is it pushing?

The answer to this question comes from Newton’s Third Law.

The way I like to phrase this law is as follows: If Object A pushes Object B, then Object B pushes Object A at exactly the same time, with exactly the same amount of force, but in the opposite direction.

So, the only reason the Earth is pushing you forwards is because you are pushing it backwards with your leg.

You might now wonder why, if you’re both pushing each other with the same force, you move forwards while the Earth stays still. Well, in fact, the Earth does accelerate backwards, but only by an incredibly tiny amount. The reason for this discrepancy comes from Newton’s Second Law. Your mass is much, much smaller than the planet’s, and so your acceleration is much, much greater when you both experience the same force. The same principle applies to sports like baseball and golf: when the club hits the ball, the very light ball goes flying away (after experiencing a very large acceleration) while the heavy club experiences very little change of motion. Or what if a bicycle and a tank were in a head-on collision? The tank would keep rolling forwards and the bike would be in pretty bad shape.

We can find innumerable other examples of Newton’s Third Law in everyday life. Sit on a rolling chair, for example, and push on something solid, like a desk. You push forwards on the desk, and the desk automatically pushes you backwards.[2. In this case, the reason the desk doesn’t move probably has less to do with the mass of the desk and more to do with frictional forces holding it in place.]

In fact, this basic principle is what moves most of the vehicles humanity has ever created. A car goes forwards because its engine spins its wheels, causing the wheels to push backwards on the road, causing the road to push the wheels forward. Airplanes and helicopters have propellers, rotors, or turbine engines, all of which push backwards on the air as they spin. And motorboats move forwards by pushing backwards against the water.

Conversely, we can use Newton’s Third Law to explain why fictional vehicles such as many of the UFOs people claim to have seen, or the flying cars or spaceships in most of science fiction, are impossible (or at least unlikely). Unless they are pushing against something else or expelling something (like burnt rocket fuel), then some other force would be needed to allow them to move or hover. This could be one of the fundamental forces. But electromagnetism requires other nearby electrical charges, harnessing the weak force involves nuclear power, and the strong force is, again, too short-range to help. Anti-gravity is the usual explanation, but at the moment it seems as though such a force probably doesn’t exist (or if it does, using it would cause a lot of problems for the ship).

romsley ufo
Except this one. This one’s real.

In other words, if you see a big metal disk zipping around in the air without any evidence that it is pushing against something, you will know that you are either watching a movie or hallucinating – according to our current understanding of the laws of physics, such movement is impossible.

Newton’s Third Law also allows us to finally explain why internal forces cannot result in motion (as discussed in the section on Inertia). Take again the example of your arm pushing forwards against your own back. We now know that while your arm pushes you forwards, your back is also pushing backwards on your arm with the same force. Since both of those two forces are actually acting on the same body (yours) and are equal in magnitude and opposite in direction, they cancel each other out and you won’t accelerate at all.

In its original form, Newton’s Third Law was stated (in Latin) as, “To every action there is an equal and opposite reaction.” Unfortunately, this phrasing has been co-opted to explain everything from basic human psychology to the folksy version of Buddhist karma. Punch someone in the nose, for example, and you’re likely to get punched back. While this may be true, it is not an accurate analogy for the pairs of forces that result from the Third Law. First of all, Third Law force pairs are always exactly equal in magnitude and exactly opposite in direction, which can’t be said of karmic retaliations. But perhaps more importantly, Third Law force pairs always occur at exactly the same time. In this sense, the word “reaction” is misleading or even incorrect. Because there is no delay between the two forces’ occurrences, neither can really be said to be “caused” by the other – rather, both are simply the result of any applied force. Lastly, while the second punch in the nose may or may not occur depending on the victim’s mood and character, no force ever exists on its own: there is always a “reaction” force along with it.

We can also use our own definition of Newton’s Third Law to identify force pairs which seem as though they might be action-reaction pairs, but actually aren’t. For example, if you’re standing outside, the Earth is pulling you downwards due to gravity and also holding you up with an equal force that acts in the opposite direction. However, this doesn’t fit into the Object A/Object B formulation of the Law. In order to find our action-reaction pairs, we could say that since the Earth pulls down on you, you must pull up on the Earth.[3. Surprising, but true – as will be explained in the chapter on gravity.] And if the Earth pushes up on you, you must be pushing down on the Earth. Now we see that there are actually four forces at work (Earth’s gravity pulls you down, your gravity pulls Earth up, Earth’s surface holds you up, your weight pushes down on Earth), all of which happen to be equal in magnitude, but for different reasons.[4. The reason the Earth pushes up on you with the same force that it’s pulling you down has to do with the nature of solids. When you stand on the ground, you are actually compressing the dirt beneath you. This compression also applies to the billions of interparticle and intermolecular electrochemical bonds in the solid matter of the Earth, which act a bit like little springs holding things together down at the atomic level. So you keep pushing down on these springs until they’re just compressed enough to hold you up against the force of gravity. If there isn’t enough solid matter to provide enough springy force to hold you up against gravity (like in mud, for example), you fall through.]

While it may be tempting to explain the reasons for Newton’s Third Law using the molecular structure of solid objects, the truth is that this is another aspect of our universe that seems not to have any reason behind it. Push on any piece of matter you can find, and it will resist your push according to Newton’s Second Law and push back on you according to the Third, all the while giving you not the slightest hint as to why it’s gotta be like that.

Big Ideas:

  • If Object A pushes Object B, then Object B pushes Object A at exactly the same time, with exactly the same amount of force, but in the opposite direction.
  • The amount of force is always the same, but it will cause different accelerations in the two objects if they have different masses (by Newton’s Second Law).
  • Cars, boats, airplanes, and humans move forwards by pushing back on the ground, the water, or the air.

Next: 2.8 – Friction and Air Resistance

Previous: 2.6 – Newton’s Second Law, Part 2: Multiple Forces