More than 2300 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 thing moved; d, the distance covered; and Δt, the time interval required. Then (1) a power equal to P will in a period of time equal to Δt move w/2 a distance 2d; or (2) it will move w/2 the given distance d in the time interval Δt/2. Also, if (3) the given power P moves the given object w a distance d/2 in time interval Δt/2, then (4) P/2 will move w/2 the given distance d in the given time interval Δt.
(a) Show that Aristotle’s proportions are included in the equation PΔt = 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.