Inequalities Applied to Kinematic Quantities

摘要

Many standard undergraduate textbooks in physics, differential equations, or mechanics contain problems in which kinematic quantities such as velocity, speed, or acceleration are expressed as functions of position. A classic example is the simple harmonic oscillator in which the velocity of the oscillating mass can be expressed in terms of its distance from equilibrium. Normally, however, when kinematics is first introduced, the relevant quantities in the initial problems are presented in terms of time, and then distance dependent problems are covered later on. There are some good reasons for this presentation order. One reason is that, for many, the distance dependent approach can be counterintuitive and confusing. Even the most brilliant minds may initially miss some of the subtleties associated with speed variations that are expressed as functions of distance. The following problem, quoted from the Einstein–Wertheimer correspondence [3], provides an illustration along these lines: An old clattery auto is to drive a stretch of 2 miles, up and down a hill. Because it is so old, it cannot drive the first mile—the ascent—faster than with an average speed of 15 miles per hour. Question: How fast does it have to drive the second mile—on going down, it can, of course, go faster—in order to obtain an average speed (for the whole distance) of 30 miles an hour?