Serway & Jewett — Conservation of Energy: Problem 36

More than 2 300 years ago, the Greek teacher Aristotle wrote the first book called Physics. Put into more precise terminology, this passage is from the end of its Section Eta: Let \(P\) be the power of an agent causing motion; \(w\), the load moved; \(d\), the distance covered; and \(\Delta t\), the time interval required. Then (1) a power equal to \(P\) will in an interval of time equal to \(\Delta t\) move \(w/2\) a distance \(2d\); or (2) it will move \(w/2\) the given distance \(d\) in the time interval \(\Delta t/2\). Also, if (3) the given power \(P\) moves the given load \(w\) a distance \(d/2\) in time interval \(\Delta t/2\), then (4) \(P/2\) will move \(w/2\) the given distance \(d\) in the given time interval \(\Delta t\). (a) Show that Aristotle’s proportions are included in the equation \(P \Delta t = bwd\), where \(b\) is a proportionality constant. (b) Show that our theory of motion includes this part of Aristotle’s theory as one special case. In particular, describe a situation in which it is true, derive the equation representing Aristotle’s proportions, and determine the proportionality constant.