Halliday, Resnick & Walker — Relativity: Problem 4

4 Apply the binomial theorem (Appendix E) to the last part of Eq. 37-52 for the kinetic energy of a particle. (a) Retain the first two terms of the expansion to show the kinetic energy in the form \(K = (\text{first term}) + (\text{second term})\). The first term is the classical expression for kinetic energy. The second term is the first-order correction to the classical expression. Assume the particle is an electron. If its speed \(v\) is \(c/20\), what is the value of (b) the classical expression and (c) the first-order correction? If the electron's speed is \(0.85c\), what is the value of (d) the classical expression and (e) the first-order correction? (f) At what speed parameter \(\beta\) does the first-order correction become \(10\%\) or greater of the classical expression?