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?