A modern Galileo tale
This applet allows for an "empirical" test of three of the most relevant theorems on the uniformly accelerated motion, as presented in the third day of Galileo's "Discourses and Mathematical Demonstrations Relating to Two New Sciences". A ball starts rolling down a "piece of wooden moulding" where a channel was cut "a little more than one finger in breadth" upon clicking the "play-pause" button. Times are measured as the ball travels certain distances and random errors are "added" to these time measures so as to simulate a real measuring process. Time uncertainty is set to s (the average human reaction time) but a different value can be chosen by typing it in the uncertainty box. The option exists of taking into account the finite size of the ball (from the point of view of energy conservation, this amounts to accounting for the rotational energy of the ball), in which case one should tick the “finite size ball” box. By default, the simulation will be carried over in the point-particle approximation, in which the appearance of the object can be further specified: a little ball so as to be consistent with Galileo’s original description, or a sliding block. Three different experimental setups are possible (labeled after the three theorems) depending on which quantities are measured and which parameters can be played with. There is also a "free mode" setup, where all parameters can be freely changed, that allows one to tailor the experiment to one's needs.
- Theorem II: In this setup the length of the ramp is fixed, but it is possible to choose where to take time measures by changing . Once the descending body has gone past the target, is displayed on screen. In this and in the following/other setups, if the finite-size-ball option has been selected, the size of the ball may be changed by choosing its radius and the cycloid traced by a point on the rim of the ball is displayed in light grey.
- Theorem IV: In this setup the height of the ramp is a free parameter, and the total fall time is measured. Time measures are displayed once the ball has rolled down the ramp.
- Theorem V: Both and are free parameters in this experimental setup, as theorem V relates the ratio of two fall times to those of the corresponding lengths and heights.
- Free mode: In this setup all parameters can be played with: , , (that is it possible to decide at which target to take time measures), and . Indeed the magnitude of gravity acceleration can be changed as well, so as to repeat the experiment on a different celestial body. Whenever the value of surface gravity is selected that corresponds to a planet in the solar system, the name of the planet is displayed on screen.