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Galileo's theorems

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:

x1 = v t1

x2 = v t2

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.

1. If a moving particle traverses two distances in equal intervals of time, these distances will be to each other in the same ratio as the speeds.
2. If two particles are moved at uniform rates, but with unequal speeds, through unequal distances, then the ratio of the time intervals taken will be the product of the ratio of the distances by the inverse ratio of the speeds.

Note to the instructor: The inspiration for this problem was the discussion of the Galileo theorems in the book Changing Minds by Andrea diSessa. It also has an extensive discussion of the role of external representation.