# How can we conduct experiments to verify formulas without special equipment? How was it done in the past?

If we wanted to verify experimentally/empirically as laymen without special instruments a formula such as $$s=16t^2$$ which is used to approximate the distance s, in feet, that an object falls freely from the rest in t seconds by using a stone for instance, how exactly would we do it?
I image that someone would need to write down the distance s the stone traveled at various times t, tabulate the results and try to derive the formula.
But what is not clear to me is how the actual experiment is done. I.e. how would one proceed to "mark" where the stone is on its falling path at various times without having special equipment? E.g. how would they do it 400 years ago?

• Tall tower with windows at multiple heights, some reasonably accurate time piece, and walking up/down a lot of stairs... Commented Mar 24, 2022 at 22:10
• @JonCuster: When doing such a test with e.g a tower of 10 story's high, how can we be sure that the values we observed will be valid if the tower went on higher? Isn't this just inductive reasoning hence not reliable?
– Jim
Commented Mar 25, 2022 at 10:18
• @Jim You make models based on the experiments and observations you can perform. You verify them with more experiments. Eventually, you get data that suggests you need a better model. So, start with the Galilean gravity we're talking about here. Newton finds that a different model fits orbits (while preserving the Galilean model as a limiting case). Later, Einstein took another step, with a completely different model that nevertheless matched Newton and Galileo within their restricted domains. The method is as reliable as the data it's based on. Commented Mar 25, 2022 at 14:57

This was done initially by "diluting" the force of gravity by rolling objects down a ramp, tilted at a shallow angle to the horizontal. Trigonometry yields the resulting (reduced) force accelerating the object. The dilution was chosen to accommodate the clocks available at the time and the experiment conducted for a range of ramp lengths.

• How were they confident that the calculations based on trigonometry gave the almost the same result as free fall?
– Jim
Commented Mar 25, 2022 at 9:02
• @ jim, arithmetic did the trick. Commented Mar 25, 2022 at 16:37
• @Jim Galileo did experiments using ramps & water-clocks. He also used singing to do timing in some of his experiments. See galileoandeinstein.phys.virginia.edu/lectures/gal_accn96.htm His reasoning was sound, but he would certainly agree that empirical testing of physical hypotheses is vital. And of course, Galileo's work predates Newton's experiments and development of mechanics and calculus. Commented Mar 25, 2022 at 16:44

Let several persons climb a tower to different heights proportional to the squares of integer numbers, $$h_n=h_1 n^2$$ (for example $$1$$ m, $$4$$ m, $$9$$ m, $$16$$ m, $$25$$ m, $$36$$ m, ...).

Then let all of them drop a stone at exactly the same moment in time. According to the law $$h=\frac{1}{2}gt^2$$ the falling times will be $$t_n=\sqrt{\frac{2h_n}{g}}=\sqrt{\frac{2n^2h_1}{g}}=n\sqrt{\frac{2h_1}{g}}$$ That means the stones are expected to hit in equal time steps of $$\sqrt{\frac{2h_1}{g}}$$.

So listen to the stones hitting the ground and check if you really hear the expected beat with equidistant time steps. Humans are very good at detecting small deviations from this regular beat.

• How does the height being proportional to the square root of integer numbers help? Wouldn't this idea work if heights were 1m, 2m, 3m etc?
– Jim
Commented Mar 25, 2022 at 9:05
• Also extrapolating that the measurements for those meters are valid for higher levels isn't it inductive reasoning and hence it does not really prove the outcome?
– Jim
Commented Mar 25, 2022 at 13:18
• @Jim Sorry, I meant to say proportional to squares, not square roots. I have corrected this and added some explanation. Commented Mar 25, 2022 at 13:22
• 1) Why did you define it as $h_n=h_1 n^2$ and not just $h_n= n^2$?
– Jim
Commented Mar 25, 2022 at 16:22
• 2) How did $h=\frac{1}{2}gt^2$ appear? The formula I was asking is different. I suppose you give an experimental approach for another example? But in this case it seems that those who perform the experiment are already aware of the formula right? But what if they weren't aware and were trying to derive it?
– Jim
Commented Mar 25, 2022 at 16:23