The parabola. Now, let me start with a fact that you may or may not know. There's only one parabola just like there's only one circle up to symmetry. There's an infinity of different ellipses, one for every value of eccentricity. There's an infinity of different hyperbolas, one for every value of eccentricity.
But the circle has zero eccentricity and the parabola has eccentricity of one. It's an ellipse where the eccentricity becomes one. And there's only one of them. So, the whole issue is one of scaling. The proposition one says, once it's a parabola, the next natural question is which parabola?
And that's a question about scale of the parabola. Parabola that goes only three feet or a parabola that goes half a mile, or in World War II they were shooting 20 ton ballistic cannon balls 22 miles. Onto the shore of islands in the Pacific. Try to picture that, okay?
And how much powder was needed to blow those things. All right, so how do we do it? Well, we do it algebraically of course, and you probably should have all seen this We describe the distance X as the initial horizontal velocity V sub zero by times the time.
That's uniform motion. And the vertical distance is one-half the acceleration of gravity times time squared. We substitute. Use the first equation to define time as x over v sub zero, we substitute n, and we get a parabola. Y is proportional, and you can see what is proportional to x square.
Galileo has to do that differently. Michael Wingate asked the question how he does it. He has to do it piece-wise, but this variation is the variation described in Apollonius and gives him a parabola. So he's perfectly correct about it being a parabola. Then, then I'm cheating on Galileo quite a bit.
I'm taking a derivative of y with respect to x to get the slope at every point. And calling that the tangent of theta. Meaning if you draw a tangent through the line equal to the slope at any point, that angle is the slope of that line and the slope of a line is a tangent.
And then the velocity at any point is the square and by way speed because it is non-directional here. Velocity at any point is the square root of the square of the initial speed plus twice g y. Why is it twice g y? Because that's the vertical component. And it's the combination of the two.
And then I give you the rest. I'm not going to through the rest of this. At impact there is a correction here. And God knows I can't find the original or I would have replaced it. I did the original maybe 15 years ago and nobody has ever pointed out there's a glaring error.
The line on the angle of impact should be two times HA and no two on the other side so I just scratched out the HA and the rest of it's fine. I just wrote something stupid down, and I only caught it this morning while I was preparing for tonight and suddenly realized that can't be right.
But the rest is correct. It tells you what the velocity of impact is which was a very big deal to the cannon people. Because their cannon balls didn't blow up when they hit. The damage was done through the impact of the ball. They didn't carry explosives. They hadn't figured out how to kill people as well as we think of them now.
But they did figure out how to blast wooden boats off, putting very big holes in wooden boats. So the velocity of impact was an important deal. Anyway, that's how we do it, and there's an obvious problem, to specify which parabola, we have to have two initial values. The initial velocity horizontally, and the acceleration of gravity.
Neither of which Galileo could get. He had almost no way of measuring muzzle velocities directly. It's still a tough measure. By the way, do people know how velocities are actually measured now out of a gun? Shoot it into a very heavy pendulum and measure the height of the pendulum.
Ballistic, it's the ballistic pendulum as it's used. And it's been used since the seventeenth century to do that for pistols. I'm not sure how you do it for a cannon ball. And we've already noticed Richie Ollie basically gives the acceleration of gravity by giving the distance in and what distance to fall in one second.
But Galileo didn't have it. So for Galileo to do this he would have. The way we do it would've had to come up with two parameters one of which is a variable. The other of which is presumably constant G is presumably constant. But V certainly V of zero is certainly.
Everybody see the problem? Because here is where he just gets extraordinarily ingenious. Saggratto raises the issue. The theory of compounding these different impedances, and of the quantity of impotence that results from such mixing, is so new to me as to leave no little confusion in my mind. I speak not of the mixing of two equable movements.
Okay, that's not a problem. I speak not of the mixing of two ethical removes, one along the horizontal line and the other along the vertical, even though unequal to one another, for as to this I quite understand that emotion results, which is equal in the square, to both components of it, that is the vector rule.
Got the square root of the sum of the squares. So he knows that for equable motion. But I am confused by the mixture of equable horizontal and naturally accelerated vertical motion. Salviati's reply, we can reason definitively about movement and their speeds or impetuses. Whether these are equable or naturally accelerated.
Only if we first determine some standard. Is the Italian word, I'll come right back to that. That we can use to measure such speeds as also some measure of time. Literally translated, if I pull out my Italian dictionary, measure. We want a measure of speeds, alternative definition, a gauge of speeds.
Okay, so Greg Bates choice of the word standard is not inappropriate, I'm the one who supposed to put measure in there. Because that's what we need, we need measures. We need a measure of speed, and we need a measure of time. And he says that almost immediately. As to the measure of time, we already have universal agreement on hours, minutes, seconds, etc.
And just as the measure of time is for us, that one in common use accepted by everybody, so it is necessary to assign some measure for speeds. To be commonly understood and accepted by all. That is, one that will be the same for everyone. Everyone at all times, right?
That's what we want for a measure of speed. As explained previously, the author deemed suitable for such a purpose the speed of naturally falling heavy bodies. Of which the growing speeds keep the same tenor everywhere in the world. That's a claim. Acceleration of gravity is everywhere the same in the world.
He has no basis for making that claim. It's almost true, but not quite. To determine and represent this unique impetus and speed, our author has found no better means. Then to make use of the impetus acquired by the moveable in a naturally accelerated motion. Okay. So here's a way to measure speeds.
The speed acquired, actually it's speed acquired in a particular fall. Particular vertical drop absent air resistance. Okay? I'll say it differently now. We're gonna use the height of fall as a proxy, proxy for what? Technically, speed squared. Now that's good, because velocity squared is a scalar, not a vector.
So at the very least, we're on something that we can become comfortable with. For those who don't understand what I just said, the square is the dot product of the lost and that's a scalar that the direction ceases to matter. Energy is a scale for that reason, that's what we're getting at here.
Okay that's the first point. Second we're saying in so far as that the speed from any height is the same in all the, everywhere in the world. We can use height as a proxy for speed square. What's an analog that you're familiar with? We use the length of a mercury column as a proxy for temperature.
Okay? Most measurement is done by proxies. This is going to be the sole way anybody measures velocity in experiments for most of the next century. Is by height of fall. Okay, so it's a very nice proposal. What's required for it to serve as a proxy? Well actually three things are required for it to be a proxy in any one experiment, the speed acquired in descent from rest is proportional to the time of descent.
Descent involves uniformly accelerated motion, that is. Second, the same speed is acquired in descent from rest from a given height regardless of the path, and the third, all bodies acquire the same speed in descent from any given height regardless of their weight, shape, etc. That's the crucial one.
We can't use height as a measure of speed If different bodies acquire different speeds, falling from the same height. So it's fundamental to using height as a measure of speed, that it's independent of the shape or weight of the body. Which I repeat, is the core assumption called the principle of equivalence in the general theory of relativity.
You're gonna see it's no less important than Newton. But this is where it's being performed. And now, for height to serve as a uniform universal measure of velocity squared and not just local measure, the increments in speed acquired in equal time have to be the same everywhere on earth.
So this is a very bold theoretical claim. Use height as a proxy for speed squared. Fair enough? But time and again, we had no way of measuring temperature until Fahrenheit came up with that crazy idea of a tube of Mercury growing. They've tried to do it with alcohol and other things.
They couldn't find a proxy. Now, we have many, many proxies for measuring temperature. We have many proxies for measuring velocity, too. We have lots of things. But this is a very bold proposal. It is absolutely good sense science in every sense of the word, and to my knowledge, it starts with Galileo making these points, cuz these are, to my knowledge, original.