When Galileo first began describing motion quantitatively, he didn't have the advantage of the use of
algebra. As a result, some of his statements were a bit hard to interpret. Here's one of his theorems: If a moving particle, carried uniformly at a constant speed, traverses two distances,
the time intervals requires are to each other in the ratio of these distances.This is in rather old-fashioned language and contains an unstated assumption.
Let's translate it into more modern terms.Consider two instances of a particle moving at a constant speed, v. If
in one case it travels a distance x1 and in the second case it travels
a distance x2, then if the times it would take to cover these two
distances are t1 and t2 respectively, then the ratio t1/t2 is
equal to the ratio x1/x2.This is quite easy to
prove using our basic definition of average velocity for a constant speed where v = <v> (velocity
= average velocity):
In our two cases, these become two equations:
If we take the ratio of these equations we get the complete modern translation:
If two particles move with the same speed v for different times, t1 and t2, then the ratio of the distances they travel, x1 and x2 is given by x1/x2 = t1/t2.
Carry out the analogous translation and proof for each of the following of Galileo's theorems into modern terms.