Free fall

Annotations by Ben Wilbrink

Physics is a broad field. In order to reach some efficiency in the study of physics and the design of physics test items, it would be nice to concentrate the effort on one or two physics topics only. The one subject that seems most suitable to this VIP treatment is that of free fall. The history of the subject, that is until Newton's laws, is available in Dijksterhuis (1924) Fall and throw. A contribution to the history of mechanics from Aristotle until Newton [original in Dutch: Val en worp]. The main results also in Dijksterhuis (1950), available in English translation (1961). An important addition to the work of Dijksterhuis concerning Galileo is Stillman Drake (1990), among others on the times-squared law of distances in fall.

The work of Galileo is indeed a turning point in the science of physics, from natural philosophy to experimental science. From the attempt to find causes of natural phenomena, to the attempt to describe them on the basis of experimental observation. From the use of philosophical language to that of mathematics to do the theorizing.

Free fall is a rather simple concept in modern physics, it should be possible for readers not knowing much or anything of physics, to follow the main examples and arguments. Yet many important epistemological problems in science—and therefore also in education—may be illustrated using free fall. The most important problem being the relation between physics and mathematics. Free fall has something to do with gravitation, yet calling gravitation the cause of free fall is name-calling only. Understanding free fall more or less stops at the description of the phenomena, the most succint description being an appropriate mathematical formula. The danger, then, is to suppose that being able to reproduce this formula or apply it in toy-situations is proof of 'understanding' free fall. Have I made myself clear? I fear not. But then the above is my program of action only.

Another example of work in physics done without any understanding of possible 'causes' of the phenomenon, is the discovery of temperature, as described by Hasok Chang in his 2008 book under this name.

Hasok Chang (2004/2007). Inventing temperature. Measurement and scientific progress. Oxford University Press.

Travis Norsen (2005). Lab 2: Galileo's Freefall Experiment. General Physics I, Marlboro College, Fall `05. [dead link? 1-2009]

E. J. Dijksterhuis (1924). Val en worp. Een bijdrage tot de geschiedenis der mechanica van Aristoteles tot Newton. Groningen: Noordhoff. [Fall and throw. A contribution to the history of mechanics from Aristotle to Newton]

E. J. Dijksterhuis (1951/1969). The mechanization of the world picture. London: Oxford University Press.

Stillman Drake (1990) Galileo: Pioneer scientist. University of Toronto Press.


Bas Haring (15 spet 2007). Bas Haring. Kennis-katern De Volkskrant, p. 7.


The distance s that a body will fall from rest in a vacuum in time t seconds is given by the formula s = ½gt2. Find its velocity after t1 seconds and its acceleration.

Solution       s = ½gt2.
Differentiating, v = ds/dt = gt.
Differentiating, α = dv/dt = d2s/dt2 = g.
Hence the velocity when t = t1 is gt1, and the acceleration is a constant g.

quoted from Claude Irwin Palmer (1924). Practical calculus for home study. McGraw-Hill. p. 93-94.

I suspect it is in itself remarkable to find free fall even mentioned in a textbook on calculus. The quoted example is more or less all that its author has to say on the subject of free fall.

A body falls from rest at a place where g = 32.2. Find (a) the velocity at the end of the third second; (b) the space fallen through in 5 seconds; (c) the space fallen through in the fifth second.

Ziwet (1893/94, Vol. 1 p. 56)

Galilei, who first discovered the laws of falling bodies, expressed them in the following form: (a) The velocities at the end of the successive seconds increase as the natural numbers; (b) the spaces described during the successive seconds incease as the odd numbers; (c) the spaces described from the beginning of the motion to the end of the successive seconds increase as the squares of the natural numbers. Prove these statements.

Ziwet (1893/94, Vol. 1 p. 56)

"Prove these statements" (box above): that is a mathematical question, not a physical one! About those 'seconds': that is an anachronism. How is it possible for a carefully formulating man like Ziwet to be so sloppy in matters historical? I doubt that Galilei used the concept of velocity [at a particular moment, i.e. in the limit], I will check on that.

A stone dropped into the vertical shaft of a mine is heard to strike the bottom after t seconds; find the depth of the shaft, if the velocity of sound be given = c. Assume t = 4 s., c = 332 metres, g = 980.

Ziwet (1893/94, Vol. 1 p. 56)

Find the velocity with whih the body arrives at the surface of the earth if it be dropped from a height equal to the earth's radius, and determine the time of falling through this height.

Ziwet (1893/94, Vol. 1 p. 60)

The above question is highly artificial, of course. It serves to exercise the somewhat involved formulas of acceleration proportional to the square of the distance (Newton's law of universal gravitation). Today one might shoot a rocket to the specified height, from which it will fall back to the earth, in 1893 such a possibility was science fiction.

Damerow, Peter, e.a. (2004). Exploring the Limits of Preclassical Mechanics: A Study of Conceptual Development In Early Modern Science; Free Fall and Compounded Motion In the Work of Descartes, Galileo and Beeckman. (Sources and Studies In the History of Mathematics and Physical Sciences) Springer.

Edwin Danson (2006). Weighing the world. The quest to measure the earth. Oxford University Press.

Alexander Ziwet (1893/4). An elementary treatise on theoretical mechanics. Part I Kinematics. Part II Introduction to dynamics; statics. Part III Kinetics Three volumes. London: Macmillan and Co.

f = ma (Newton)

Force equals mass times acceleration. Is this classical formula (a) a law of physics, (b) the definition of force, (c) an axiom formulated by Newton?

Henk J. M. Bos: De zeventiende eeuw — wiskunde aan het begin van de Moderne Tijd. In Machiel Keestra (Red.) (2006). Een cultuurgeschiedenis van de wiskunde. Uitgeverij Nieuwezijds.

Norwood Russell Hanson (1965). Newton's First Law: A Philosopher's Door into Natural Philosophy. In R. G. Colodny. Beyond the edge of certainty. Essays in contemporary science and philosophy (pp.6-28). University Presss of America.

Brian Ellis (1965). The Origin and Nature of Newton's Laws of Motion. In R. G. Colodny. Beyond the edge of certainty. Essays in contemporary science and philosophy (pp. 29-68). University Press of America.

"An impressed force is an action exerted upon a body, in order to change its state, either of rest, or of uniform motion in a right line."

Newton, Principles, p. 2, as translated in Jammer 1957/1999 p. 121.

"Since newton clearly distinguishes between definitions and axioms (or laws of motion), it is obvious that the second law of motion was not intended by Newton as a definition of force, although it is sometimes interpreted as such by modern writers on the foundations of physics. Nor was it meant to be merely the statement of a method of measuring forces. Force, for Newton, was a concept given a priori, intuitively, and ultimately in analogy to human muscular force. Definition IV, therefore, is not to be interpreted as a nominal definition, but as summarizing the characteristic property of forces to determine accelerations."

Jammer 1957/1999, p. 124

Test items

  1. A straight tunnel is bored through the earth to connect two points A and B on the surface. Show that under certain assumptions, which you should state, the time T to fall freely from A to B is independent of A and B. Caluculate T in minutes, taking the earth's radius to be 64000 km.
  2. Comment on the feasibility of a web of such tunnels as a global transportation system.
  3. Does a straight tunnel provide the quickest connection between A and B? Discuss briefly.

Thompson 1990, p. 5, #11. The answers: p. 51-52 (click here for the answers to 1. and 2.).


simulation applet For an applet simulating the Galileo setup of rolling balls, see problem number one here: html, click on the link to the Java applet. The applet allows editing of parameters of the experiment. The course was given in 1999 by Larry Gladney, professor of Physics and Astronomy (see his home site)


Nancy J. Nersessian (1992). How do scientists think? Capturing the dynamics of conceptual change in science. In R. Giere: Cognitive models of science. University of Minnesota Press. pdf

reardon et al figure 1a Patrick T. Reardon, Alan L. Graham, Shihai Feng, Vibha Chawla, Rahul S. Admuthe and Lisa A. Mondy (2007). Non-Newtonian end effects in falling ball viscometry of concentrated suspensions. Rheologica Acta, 46, 413-424. pdf

Aant Elzinga (1972). On a research program in early modern physics. Götegorg: Akademieförlaget.

Gerrits, G. C. Gerrits (1939, 1941). Leerboek der natuurkunde. Brill. – 3 delen, linnen, 22e, 18e, 17e druk resp., 249 + 227+254 blz, ingevoegd overzicht deel II 27 blz, idem deel III 24 blz.

M. Minnaert (1971 (3)). De natuurkunde van 't vrije veld. 3. Rust en beweging. Thieme.

C. D. Andriesse (1993/2007). Titan kan niet slapen. Een biografie van Christiaan Huygens. Olympus.

July 16, 2010 \ contact ben at at at    

Valid HTML 4.01!