Could You Have Defended Galileo?
The Problem. When Galileo died in 1642, there were still two competing schools of free fall. The Galilean School upheld Galileo’s law of free fall, which may be posed either of two forms:
- d ∝ t2, or
- v ∝ t,
for any freely falling body. The first claim is the famous time-squared law. The second claim says that velocity in free fall is proportional to time fallen. It’s easy to prove that these two claims are equivalent.
On the other hand, the Jesuit School upheld that
- v ∝ d.
That is, they believed that velocity in free fall is proportional to distance fallen, which was the traditional view at the time. I call this the ‘Jesuit law’ of free fall.
There was no agreed upon experimental evidence in the 1640’s that could verify one law and falsify the other. (Each side claimed to have experiments that vindicated their law, and disproved their competitor’s law.) However, a clever theoretician might still try to use purely theoretical means to prove one side false.
The Challenge. Prove that the Jesuit law is false, without assuming Galileo’s law.
Some Remarks Before You Get Going. These aren’t really hints, just a few comments about what you should and shouldn’t take for granted.
- You may assume that for uniformly moving objects, d = vt. This law is as old as Aristotle, and was agreed upon by both schools of free fall.
- You may assume any mathematics known to Euclid or Archimedes.
- The modern calculus will lead you astray. (It’s too easy to accidently assume something that implies Galileo’s law when you start using these tools. If you assume Galileo’s law, you haven’t answered the challenge. And anyway, nobody knew what a derivative was in the 1640’s, so that’s kind of cheating.)
- I’ll sketch a solution to this challenge next week.
A Hint. (Warning! Stop reading if you don’t want any hints!) Remember, you go into this knowing almost nothing about the way bodies fall. But you can still try to put upper and lower bounds on motion in free fall, in order to get your result. How can the law of uniform velocity provide bounds on the time it takes for a body to fall?
On Monday, I’ll elaborate on that last hint. (It’s actually an interesting problem all by itself.) Later, I’ll sketch a little bit of the first answer to this challenge, which was given by Pierre de Fermat in an obscure letter to Gassendi, around 1646.
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Are we allowed to assume that free falling bodies move faster than uniformly moving bodies?
Well, that’s not in general true.
What you can (and should!) asssume is this. Choose any two points A (higher) and B (lower) in the path of a falling body. Suppose the falling body has velocity v_A at A and v_B at B. Then anything that moves with uniform velocity v_A from A to B will travel no faster than the falling body on [A, B]. Similarly, a body that travels with uniform velocity v_B from A to B will travel no slower than the falling body on [A, B].
This is (the first step of) how you can use uniform velocity to put bounds one motion in freefall.
Yeah, that’s what I meant. Sloppy phrasing on my part.