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You can write s = ½at² in one line. Galileo had no algebra of that kind, no stopwatch and no way to watch a falling stone closely enough to measure it. What he did instead, roll a ball down a slope and time it with water, is an early and famous example of careful measurement being used to test an idea about motion.
What exactly did Galileo do?
The experiment is described in *Dialogues Concerning Two New Sciences*, published at Leyden in 1638. In the English translation by Henry Crew and Alfonso de Salvio (1914), Galileo's spokesman Salviati says they took "a piece of wooden moulding or scantling, about 12 cubits long, half a cubit wide, and three finger-breadths thick", cut a straight groove in its edge, lined it with polished parchment, and rolled "a hard, smooth, and very round bronze ball" down it, with one end raised one or two cubits (Two New Sciences, Crew and de Salvio translation, Internet Archive).
Timing was the hard part. They placed "a large vessel of water" up high with a thin pipe at the bottom, collected the water that ran out during each descent in a small glass, and weighed it "on a very accurate balance". The ratios of the weights gave the ratios of the times. Repeated runs, the text says, never differed by more than "one-tenth of a pulse-beat".
What did he find?
Rolling the ball a quarter of the way down took "precisely one-half" the time of the full run. Trying other fractions, "in such experiments, repeated a full hundred times, we always found that the spaces traversed were to each other as the squares of the times, and this was true for all inclinations of the plane."
In the same book Galileo calls a motion uniformly accelerated "when starting from rest, it acquires, during equal time-intervals, equal increments of speed", and states the result as a theorem: distances covered from rest "are to each other as the squares of the time-intervals". He also gives the pattern in another form: distances covered in equal successive intervals "stand to one another in the same ratio as the odd numbers beginning with unity."
Why use a slope at all?
A gentle slope stretches the motion out. A stone dropped from head height lands in well under a second, far too fast to time with water. On a board raised one or two cubits over twelve, the ball takes long enough for the water clock to collect a weighable amount. Because the squares-of-times rule held at every inclination, it was reasonable to expect it to hold as the slope approached vertical, that is, for free fall.
The slope also led Galileo towards inertia. Later in the book he writes that "any velocity once imparted to a moving body will be rigidly maintained as long as the external causes of acceleration or retardation are removed, a condition which is found only on horizontal planes." The NCERT Class 11 textbook retells this through his double inclined plane: a ball rolling down one smooth plane climbs nearly to the same height on the other, and as the second plane is made flatter it travels farther, until on a horizontal plane its motion would never cease if there were no friction (NCERT Class 11 Physics, Chapter 4). Newton later credited Galileo in the *Principia*: using the first two laws, he wrote, "Galileo discovered that the descent of bodies observed the duplicate ratio of the time" (Principia, Motte translation, Wikisource). We compare Newton's own wording with the textbook version in what Newton's three laws actually said.
Try a home version in 20 minutes
You need a long, straight plank or a curtain rail, a marble or steel ball, a book to raise one end, tape and a phone stopwatch.
Mark the start, then the points at 1/4, 1/2 and the full length.
Predict first: if distance goes as time squared, the 1/4 mark should take half the full time, and the 1/2 mark about 0.71 of it.
Time each distance five times and average. Human reaction time adds error, so keep the slope gentle so that runs last a couple of seconds.
Divide each average time by the full-length time and compare with your prediction.
Raise the slope and repeat. The times change; the ratios should not.
If your ratios drift, ask why before blaming the theory: is the ball slipping, is the rail bent, is your thumb late? That habit, checking the apparatus before doubting the law, is the real lesson of Galileo's board. Our guide to reading physics graphs shows how to plot your results so a straight line tests the rule.
If motion is easier for you to understand when you can see it, the Fermi Simulation Lab has interactive simulations for ideas that are easier to see than to read. Make your prediction before you press play.
