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Episode 2: The Law Of Falling Bodies - The Mechanical Universe – Transcript
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- In a vacuum, all bodies fall with the same constant acceleration regardless of their weight.
- Galileo theorized that in successive intervals of time, the distances fallen follow the odd numbers.
- Isaac Newton and Gottfried von Leibniz discovered differential calculus, which introduced the derivative to describe motion.
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It was discovered by Galileo, refined by Isaac Newton, and in the hands of Albert Einstein provided a theory of the mechanics of the cosmos.
It was one of the deepest mysteries in all of physics. All bodies fall with the same constant acceleration. In a vacuum, all bodies fall with the same constant acceleration.
That's it. That's the law of falling bodies. It doesn't seem like much to get excited about, and yet just look at what it says. First of all, it says that the effect of gravity on all bodies is the same regardless of their weight.
From Galileo to Isaac Newton, right down to Albert Einstein, that's been one of the central mysteries in all of physics. Furthermore, it says that bodies fall with constant acceleration.
It's almost impossible even to understand what that means without a marvelous mathematical device called the derivative. We'll see today what that means.
And finally, profound and important the law this is, it violates our simplest intuition because it happens only in a vacuum, not in the world we're familiar with. For all of us, the effect of the Earth's gravity was probably our first encounter with the laws of nature.
And whether or not we understand how gravity works, we have an innate fear of what it does. But exactly what is the effect of gravity? Some bodies fall to the Earth quickly and directly, while others behave quite differently.
In some cases, the pull of gravity can be resisted almost indefinitely. To make any sense at all about how and why bodies fall, we need to separate the effect of gravity on a falling body from the opposing effect of the air through which the body is falling.
In other words, we have to imagine a body falling not through the air, but through a vacuum. For instance, what happens if a penny and feather drop simultaneously from the same height?
They behave exactly as we'd expect, each falling at a very different rate than the other. But that's only because of the effect of air resistance on the two objects. In a vacuum, a penny, a feather, and any other object will fall at exactly the same rate.
With virtually no air remaining inside the glass tube, the penny and feather are now in a vacuum. When the penny and feather are released, we'll witness the law of falling bodies in action. Without the effect of air resistance, in other words, in a vacuum, all bodies, regardless of their weight, will fall at exactly the same rate.
What could be a better When Apollo 15 astronaut David Scott explored the airless surface of the moon, he couldn't resist repeating this classic experiment for all the world to see. Well, in my left hand I have a a feather. In my right hand a hammer.
And I guess one of the reasons uh we got here today was because of a gentleman named Galileo a long time ago who made a rather significant discovery about falling objects in gravity fields. And we thought that uh where would be a better place to confirm his uh findings than on the moon.
And I'll drop the two of them here and hopefully they'll hit the ground at the same time. How about that? Well, I'll be darned. It looks as if Mr. Galileo was correct in his findings. Mr. Galileo was correct.
Nearly 400 years ago at a time when all the world believed that heavy bodies fall faster than lighter ones, Galileo realized that in a vacuum all bodies should fall at the same rate. Galileo couldn't produce a vacuum, but he could imagine one.
He pictured a heavy body attached to a lighter one. With this compound body he asked, "Will it fall faster or slower than the heavy body alone?" If the lighter body did fall more slowly, it should slow down the heavy body.
So, the compound body should fall more slowly than the heavy body alone. But, the compound body is actually heavier than the heavy body alone. Therefore, the compound body should fall faster than the heavy body, not slower.
Obviously, the long-held view that the heavier a body is, the faster it falls leads to an inescapable contradiction. Galileo realized that the only logically acceptable view was that all bodies, regardless of their weight,
fall at exactly the same rate once the effect of air resistance is removed. Of course, if all bodies in a vacuum fall at the same rate, the next question is exactly what is that rate?
From common experience, we do know one thing about the rate of a falling body. The speed of a falling body increases as it falls, which means that it accelerates, dropping faster and faster as it falls.
Even before Galileo, a number of scholars tried to formulate a description of this accelerated motion. Some 100 years earlier, Leonardo da Vinci made his own study of falling bodies, driven perhaps by his dream of human flight.
Rather than ask how fast a body was falling, da Vinci asked how far would it fall in successive intervals of time. His theory of accelerated motion was that a body would fall greater distances in later intervals.
He theorized that those distances would follow the integers. That is, one unit of distance in the first time interval, two units of distance in the second time interval, and so on. Galileo himself adopted da Vinci's method of description, but he reached a different conclusion on how the distance increased.
Instead of increasing as the integers, Galileo's theory was that in successive intervals of time, the distances should follow the odd numbers. Falling one unit of distance in the first time interval, three units of distance in the second interval, five units of distance in the third interval, and so on.
In other words, according to Galileo, the distance fallen is proportional to the odd numbers. Galileo reached his conclusions after a brilliant series of experiments in which he timed a ball as it rolled down steeper and steeper inclines, moving closer and closer to the vertical path of a freely falling body.
Galileo's law of odd numbers can be seen in action in a very unlikely place. It's a place that would have amazed that great Renaissance thinker even more than the surface of the moon. At Magic Mountain Amusement Park in Southern California, customers gladly pay for the privilege of plummeting through space under the influence of gravity.
I'll buy you one. Some cotton candy. We can feed the ducks. No. I don't think I want to do this. Oh, no. Actually, that part of the ride is free. Here we go.
What the customers have really paid for Okay. is an arrangement that allows them to survive. At any rate, what about Galileo?
If this is one unit distance, this should be three. This should be five. And so on. Which is exactly what they are. Galileo was right. In successive intervals of time, the distances fallen do follow the odd numbers.