infinite jerk

Last week I worked through some kinematics, and it led to a discussion in lecture about the jerk, that is, the time derivative of the acceleration. I was emphasizing that the velocity must be a continuous function of time (lest we have infinite accelerations and hence infinite forces), but then I realized that most people in the class (physics majors) felt that the acceleration must also be a continuous function of time.

After a bit of thought I agreed with them, but not for the reasons they wanted; indeed several were unsatisfied with my discussion: As far as I can tell, the only thing that limits the jerk is the propagation of information. When you slam on the brakes on your car, it takes a finite time for the brake shoe to come in contact with the wheel. This is not really a fundamental problem with infinite jerk (the jerk appears in no physical law), but in any real situation because information propagation (and other kinds of changes) happen at finite speeds, the jerk can never really be infinite.

vector subleties

It is vectors all week this week, in my class, and in two classes I have taught for others. It is understandable that they confuse students, even physics majors with good backgrounds. Here are some subleties that I like to point out:

• Vectors have a magnitude and a direction, but that is not sufficient. They also have a coordinate-free existence or description, and they form a linear space (with the usual linear operators). In this sense, despite what every textbook says, the unit vectors that define the coordinate system are not vectors!
• Although vectors carry around all this geometric baggage, they have a magnitude and a direction and nothing else. I can still confuse the physics majors by sliding around vectors on the board. There is no position associated with a velocity vector, and we confuse the students by always drawing the velocity as coming from the object that is moving.
• Multiplication of a vector by a scalar is usually conceived as changing the magnitude of the vector, which it does, but it also changes the units, in many cases of interest (for example when a displacement is multiplied by an inverse time to make a velocity). So it often produces a new vector that is not longer than the original vector, nor shorter, but really incomparable.
• There is a perfect symmetry between the relationship between velocity and position and the relationship between acceleration and velocity. However, it is far harder for students to understand that the acceleration vector can point perpendicular to the velocity vector than it is to understand that the velocity vector can point perpendicular to the position vector. No amount of class time spent on this point is wasted, in my experience.

everything is an approximation

One of the main things I emphasized in today's class (computing a trajectory in gravity near the surface of the Earth with no air forces) is that every calculation in physics is an approximation. The parabolic trajectory near the surface of the Earth is an approximation to the tip of a very eccentric ellipse, and the eccentric ellipse comes out only in the Newtonian approximation to GR, and even that only holds if there are no other forces acting (and there always are). There are also small adjustments for reduced mass, and if the object has non-trivial extension. Crazy! And in high school this is all taught like it is exact: Just plug numbers into the equations!

dimensions and symmetry

In class, I spend a lot of time on the dimensions of physical quantities. If the left-hand side of an equation is an energy, then the right-hand side must be also. Or if the left-hand side is a quantity measured in J, then the right-hand side must be also. Is it right to call this property a symmetry of physical law? It acts very much like a symmetry, because it selects out of all the things you might write down a very small fraction that could conceivably be physical laws or physical results. On the other hand, these issues are so fundamental, they almost transcend that.