Some superifical observations have been made as, for instance, that the free motion of a heavy falling body is continuously accelerated. But to just what extent this acceleration occurs has not yet been announced. For so far as I know, no one has yet pointed out the distances traversed during equal intervals of time by a body falling from rest stand to one another in the same ratio as the odd numbers beginning with unity.
It struck me that this is easy to check using nothing more than high school science. All that is required is one of the three basic equations of linear motion:
s = ut + ½at²
The results work out like this:
- In the first second, the “heavy falling body” travels 5 metres
- In the second, it travels an additional 15 metres for a total of 20 metres
- In the third, an additional 25 metres for a total of 45 metres
- In the fourth, an additional 35 metres for a total of 80 metres
The numbers 5, 15, 25, 35, etc stand in the same ratio to each other as 1, 3, 5, 7, etc. Galileo, writing in a time before the formulation of this equation, was right!
I’m continually amazed by the mental agility of the great minds of the past. There’s something to be said for basic, applied knowledge as opposed to the more abstract knowledge we’re taught and the so-called “effort saving” tools (like calculators) which only serve to make us lazier.
Galileo’s observation has enormous practical application. For instance, it gives us a neat way of estimating how deep the well is. You may need to know this if you’re ever trapped in the cellar of a haunted house with the foul creatures of the night hammering on the makeshift barricaded door. What? Why? Because the well is the only potential escape route. Don’t you watch movies?
Count the number of seconds (n) taken for the stone to hit the water at the bottom of the well. Add up the first n odd numbers starting at 1. Multiply the result by 5 metres. For example, the stone takes three seconds to fall. That means the water is 1+3+5=9*5m or 45 metres down the shaft.
(If you really want to get fancy, you could just count the number of seconds taken for the stone to hit the water, square it then multiply by 5m. But that’s another story.)