Monday, October 16, 2006


In AP Physics C, we're not afraid of calculus. So when we study kinematics, we're not restricted to motion with constant acceleration. We can handle changing acceleration! We can just integrate it to find velocity, and integrate again to find position, so who cares whether it's constant or not? Yeah! We're not tied to four plug-and-chug equations anymore.

So we've had fun looking at the properties of jerk. Yes, that's really the accepted term for the derivative of acceleration (the third derivative of position). There are as yet no official terms for higher derivatives, but snap, crackle, and pop have been proposed for the 4th, 5th, and 6th derivatives of position. I wouldn't be surprised if these names get canonized one day; as whimsical as they are, they're no less serious than up, down, charm, strange, top, and bottom.

I explained why jerk was called jerk. If you're moving at constant velocity, it doesn't feel like anything, since there's an inertial frame of reference in which you're not moving at all. If you're moving with constant acceleration, it feels like you're experiencing a constant force, which feels just like a gravitational field, thanks to the equivalence principle. If you're moving with non-constant acceleration, then you experience a non-constant force. Since this "force" is changing, you feel jerked around.

One student asked "What would snap feel like?" I didn't have a great answer, so I said "One second there's no jerk, and then the next second there is. That's snap!" Another student said "So if you want to call someone a jerk, then when they walk into the room, you can say 'Oh snap!'" My children have defeated me.

I gave a practice problem where acceleration was given as a function of time, and included a term proportional to the square root of time. They had to find the velocity, position, etc. A student astutely pointed out that this means that the jerk is undefined at t=0. Readers, you can correct me if I'm wrong, but I said that's totally legal, and here was an intrusion of actual physics into our otherwise purely mathematical world of kinematics. Newton's Second Law says F=ma, so the acceleration must exist and be finite at each point in time. Acceleration can't be infinite, because this would require an infinite force, and that's impossible. (You might say that an inverse-square force is infinite when r=0, but I would respond that classical electromagnetism doesn't really allow point charges, but only finite-density charge distributions. This probably means we should be a little bit more careful about using point charges in all our examples, but they're useful approximations if we don't think too hard about it. But point charges can't exist because they would result in infinite electric fields, which results in infinite energy density, and if you integrate the energy over any finite volume containing a point charge, you get infinite energy. If you were to define the energy density of the gravitational field, I think you would get a similar result showing that point masses are impossible.) Therefore, velocity must be continuous. However, there's no reason that acceleration needs to be continuous (and thus no reason that jerk always needs to be defined), since it's theoretically possible to have a force that's there one instant and gone the next.

This gets into an argument that I used to have with a former colleague, MF. He would claim that Newton's Third Law is the only one of Newton's laws that contains any actual physics. (I like teaching my students "Newton's Zeroth Law", an example of how physics education research can provide insights not only about physics education, but about physics. But that's not really one of Newton's laws.) Of course, Newton's First Law doesn't tell us anything that we couldn't easily deduce from the Second Law by setting force to zero, assuming nonzero mass, and solving for acceleration. It's worth teaching anyway, not only for historical reasons, but because this specific case is contrary to the intuitive understanding of how forces work (just ask Aristotle), and it provides a useful way of thinking about forces in equilibrium. But in a pinch, we could get by without the First Law. (Similarly, the chemistry teachers in my office complain about having to teach Boyle's, Charles's, and Gay-Lussac's Laws when they can all be subsumed under the Ideal Gas Law, so the students have to learn 4 laws (again, perhaps primarily for historical reasons) when one would suffice. Giving the other three laws their own names might help in keeping track of the various direct and inverse relationships, but in this case, the extra laws might just be more confusing. I don't know, not my field. Except when the freshmen say they want to take the Physics SAT II.)

So really, the dispute is about the Second Law. Does it say anything meaningful about physics, or is it just a definition of force?

MF follows Feynman, who says (Feynman Lectures, volume I, chapter 12):

If we have discovered a fundamental law, which asserts that the force is equal to the mass times the acceleration, and then define the force to be the mass times the acceleration, we have found out nothing. We could also define force to mean that a moving object with no force acting on it continues to move with constant velocity in a straight line. If we then observe an object not moving in a straight line with a constant velocity, we might say that there is a force on it. Now such things certainly cannot be the content of physics, because they are definitions going in a circle. The Newtonian statement above, however, seems to be a most precise definition of force, and one that appeals to the mathematician; nevertheless, it is completely useless, because no prediction whatsoever can be made from a definition.

However, I would respond by also quoting Feynman (the very next paragraph):

For example, if we were to choose to say that an object left to itself keeps its position and does not move, then when we see something drifting, we could say that must be due to a "gorce" -- a gorce is the rate of change of position. Now we have a wonderful new law, everything stands still except when a gorce is acting. You see, that would be analogous to the above definition of force, and it would contain no information. The real content of Newton's laws is this: that the force is supposed to have some independent properties, in addition to the law F = ma; but the specific independent properties that the force has were not completely described by Newton or by anybody else, and therefore the physical law F = ma is an incomplete law. It implies that if we study the mass times the acceleration and call the product the force, i.e., if we study the characteristics of force as a program of interest, then we shall find that forces have some simplicity; the law is a good program for analyzing nature, it is a suggestion that the forces will be simple.


For example, in dealing with force the tacit assumption is always made that the force is equal to zero unless some physical body is present, that if we find a force that is not equal to zero we also find something in the neighborhood that is a source of the force. This assumption is entirely different from the case of the "gorce" that we introduced above. One of the most important characteristics of force is that it has a material origin, and this is not just a definition.

A more concise version of this (from chapter 9):

In order to use Newton's laws, we have to have some formula for the force; these laws say pay attention to the forces. If an object is accelerating, some agency is at work; find it.

As we currently understand it, the universe has four fundamental interactions. We'll ignore the weak and the strong interactions in this post, because we're talking about classical physics, and they don't operate on large enough scales to be adequately described by classical physics. That leaves the gravitational and electromagnetic interactions, and as far as we are aware, they are forces; i.e. they cause objects to have acceleration: the second derivative of position, not the first, third, fourth, etc. So Newton's Second Law is actually making a physical statement about our universe, since it says that the fundamental interactions of our universe (which are not themselves fully described by Newton's Laws, even in the classical approximation) cause acceleration.

We could imagine alternate universes where these fundamental interactions cause velocity, jerk, snap, or something else entirely (e.g. a universe where the components of motion are not independent, so force cannot be described by a vector quantity).

Therefore, I gave the AP class this extra credit question on their most recent test. Some of them even got it right! Bonus points to the first Mah Rabu reader who can solve it. Post your answers in the comments.

Consider an alternate universe where Newton’s Second Law is F = mj. In our universe, all objects in free fall in a uniform gravitational field have a constant acceleration. In this other universe, they would have a constant jerk instead! In our universe, a projectile thrown from a level surface at a given initial speed would have the greatest horizontal displacement if it is thrown at an angle of 45° from the horizontal. What is the corresponding angle in this alternate universe? (You can assume that the projectile has no initial acceleration.)


  1. I get .615 radians, or about 35 degrees.

    Now I'm going to go out on a limb. I agree that inverse square forces aren't defined for r=0, but I think I once took a class wherein we modeled point charges with delta functions, which have the expected behavior when integrated. Sadly, I don't recall what we achieved by doing so.


  2. I get .615 radians, or about 35 degrees.

    Correct!!!! You win! Also known as sin^-1 (1/sqrt(3)).

    More generally, if F = m*(d^n)x/(dx)^n, then the answer to this problem is sin^-1 (1/sqrt(n)).

    E.g. if n=2, then it's sin^-1 (1/sqrt(2)) = 45 degrees.
    If n=1, then the projectile never gets off the ground, since it can't maintain an upward velocity for more than an instant.

    Now I'm going to go out on a limb. I agree that inverse square forces aren't defined for r=0, but I think I once took a class wherein we modeled point charges with delta functions, which have the expected behavior when integrated. Sadly, I don't recall what we achieved by doing so.

    The problem isn't with defining the force at r=0. Infinite charge density is no problem, since you can integrate the charge density over some volume and get a finite charge. Likewise, Gauss's Law, etc., should work just fine. The problem is with getting infinite energy. But maybe I did something wrong.

  3. I think I prefer the posts where you incite religious radicals :-)