Okay this may not be the most interesting thing to think about, but it’s a question I’ve been asked on several occasions. For that reason alone, I started thinking about it. I also think it’s another situation where coordinate-free vector manipulation can simplify otherwise messy questions and problems. So, here’s the question.
Give three different derivations of the classical expression for a particle’s kinetic energy and show explicitly where the factor of one half comes from in each derivation. For the first derivation, assume a standard Cartesian coordinate system with the particle moving along the x-axis (or y-axis, or z-axis…adjust your notation accordingly). Start with the most general definition of work, and then you may use the traditional kinematic equations that assume constant force. For the second derivation, begin with the relativistic expression for a particle’s total energy. Make no assumption about a particular coordinate system (you may need to justify to yourself why you can do this (the Lorentz factor holds that secret). For the third derivation, again start with the most general definition of work and proceed using ONLY vector manipulation. Do not invoke a coordinate system, and make no assumption about constant force. You may need to first work out the expression for the derivative of the dot product (I HATE that name!) of two vectors. Remember that your main goal is to explicitly show where the factor of one half comes from. For each derivation, write one complete sentence describing where the one half comes from.
I’m particularly fond of Feynman’s point that (and I can’t find the specific quote…I’m paraphrasing big time here) one way to know if you really understand something is to approach it from different sides and with different methods and see that you get the same result. I want my students to be able to do this, even for something as seemingly esoteric as this. Esoterica, though, may be good for people with minds that wander and who knows where that wandering may take them?
This post is inspired by the October 2015 AstroNotes in The Physics Teacher. I have sometimes introduced vectors into my introductory astronomy course and students were able to do most of the things described below. We never discussed angular momentum or the Laplace-Runge-Lenz vector, but the other quantities were familiar. I was not permitted in my article to comment on whether or not, and to what extent, student found this work helpful because there is no published study on this topic (i.e. vectors in introductory astronomy, something that most would say is mathematical taboo) and because I could produce no hard evidence. These restrictions were quite disturbing given that I made no extraordinary claims to begin with. Such is the world of publishing. Nor was I permitted to go into details of how the geometric properties of vectors encode a lot of physics without the need for coordinate systems and the problems with traditional approaches to this topic. Such is the world of publishing.
Consider a standard two-body gravitational interaction with one object orbiting the other. Treat the mass of the orbiting object as less than the mass of the other object. For both a circular orbit and an elliptical orbit, draw an appropriate diagram showing both objects, the orbital path (assume counterclockwise orbital motion), the orbiting object at four randomly chosen points on the orbit, and each of the following vector quantities appropriately placed and approximately scaled relative to each other. Use a different style of arrow for each quantity.
- the orbiting object’s position relative to the other object
- the gravitational force on the orbiting object
- the orbiting object’s momentum
- the orbiting object’s angular momentum relative to the other object
- the Laplace-Runge-Lenz vector (sometimes called the eccentricity vector)
- the torque on the orbiting object
At each of those four points, using only the geometric properties of vectors and no numerical calculations, describe what is happening to the orbiting object’s kinetic energy and what is happening to the two-body system’s gravitational potential energy. Compare and contrast the differences in these descriptions for the two orbits.