ï»¿ >> Hello, class, Professor Anderson here. Let's take a look at the following problem. Let's say we're standing at the top of the Empire State Building, and we're going to throw a baseball off the building horizontally, and we want to see how far it will go. Okay; so let's draw the picture here. Here we are standing on top of the Empire State Building. We've got a baseball; we are going to launch it horizontally. And now it, of course, follows this parabolic trajectory, and eventually it's going to hit the ground. And let's ask the question, "How far does it go?" Okay? Well, to answer that question, we first need to decide on a coordinate system. So let's make our origin there right at the bottom of the building; X initial of zero. X final is what we're looking for. Y final is the ground, so that's zero. Y initial is up there where we started, height H. Okay; let's take a guess. How many of you have been to the top of the Empire State Building? Okay. Now, you've been up there and you've looked off. How far do you think you could possibly throw a baseball? Who has a thought? Let's just take a guess. How far will it go? Somebody raise their hand and give me a guess. >> About a thousand feet? >> Okay, a thousand feet. All right; how did you come up with that number? >> Well, the Empire State Building is roughly about 400 meters, about 1200 feet. >> Yes. >> So if it's a parabolic motion, it might go about the same height as it is. >> Okay. So you said, "Well, if it's about that high, we can probably throw it about that far." That seems like a reasonable guess. Anybody else agree or disagree with that? Okay; let's use that as a working hypothesis. I like it a lot. Now, let's see, where do we want to go next? We want to calculate how far it's going to go, which means we're looking for X final. How do we calculate X final? Well, we just go back to our kinematic equations. Right; what we said last time was the kinematic equations apply, and we can write down equations like this, "Y final equals Y initial, plus VY initial times T, plus one-half AY T-squared." Okay; that one looks pretty good. But that doesn't have an X final in it; that only has Ys in it. But that doesn't have an X final in it; that only has Ys in it. So that probably is not going to help us. Let's go to the X equation. X final equals X initial, plus VX initial times T, plus one-half A sub X, T-squared. That, of course, has X final in it. And that's what we're looking for. That's good. X initial, we know; we started at zero. VX initial, we would have to give you that. So we're going to say that we do know VX initial. Time we don't really know yet. Acceleration, we do know, right, that's equal to zero. So this is one equation, but we've got two unknowns. And that's a big no-no. Right; if we have two unknowns, we need two equations. All right. But look, we have another equation right here. Maybe that equation can help us. In fact, it seems like we might be able to solve this equation for time, and put it right into this equation over here. And we've already done that. Right; we have solved this equation for time already, and what we got was T was equal to the square root of 2H over G. Aha, so now I can just plug all that into this equation right here. X final equals -- X initial we said was zero. VX initial we give you. And T is now square root 2H over G. A sub X is zero. That whole term goes away. And so now we have our solution. X final equals VX initial, square root 2H over G. Now, we took a guess of a thousand feet. Why did we guess a thousand feet, because we said, "Well, the building is maybe 1,000 feet high, so maybe we can throw it a thousand feet this way." So are you a baseball player by any chance? >> Used to be. >> Used to be; all right. What was your fastest throw that you've ever been able to do? >> About 55 miles an hour? >> 55 miles an hour. Okay; that's pretty fast, actually. Fifty-five miles per hour. We are, of course, going to work in SI units. So how fast is that in meters per second roughly? Anybody remember the conversion? >> Isn't it like one-half? >> Yes, it's one-half, right? >> So 22 [overlapping]? >> If you have miles per hour, cut it in half, and that's going to give you meters per second, all right? So half of 55 is 27-1/2, but we'll just say that's 27. Okay; and let's use that as our number. So we've got 27. And now we're all in SI units, so we don't have to write the units down every time. We've got a two, we've got the height of 400, and we have G of 9.8. And somebody run that in their calculator and tell me what you get. I'll approximate it here. We knew that the time was about 9; 27 times 9 is, what, well that's pretty close to 30 times 9, which is 270. And then we need to subtract a little bit, so maybe 250. Did somebody get a number? What did you get for your number? >> Two hundred and forty-three point nine. >> Two hundred and forty-four meters. Let me move that over a little bit; you can't quite see it on the screen. We'll put it up here. X final is 244 meters. Okay; how many feet is that, roughly? Well, it's roughly three feet to a meter, right, it's actually about 39 inches to a meter. But if we said three, then this would be 250 times 3 is about 750 feet. Okay; so our guess of 1,000 feet was pretty good. Right; if you had guessed 1,000 feet and you threw it 750 feet, that's very close, right; you're 75% of the way there. So that seems like a pretty reasonable guess, and that seems like a pretty long throw. Right; if you can throw something 750 feet, that's pretty significant. All right. Obviously, if you throw it faster, or if the building's higher, then it's going to end up going further. And you can see that right here in this final equation. And this is one of the reasons that I want you to calculate all these things with variables the whole time, to make sure at the end it makes sense. If I increase my speed, XF goes up. That's what that equation tells me. If I increase the height, I increase the distance. If gravity is suddenly stronger, then I would decrease the distance. Okay; and so there's a lot of information right here in this thing. And in fact, if I thought about doing this experiment not on the Earth but on the moon, could I throw that baseball further than on the Earth, or not as far? Who's got a thought of this; maybe over here on the left? >> Ignoring gravity would make the ball go further. >> Okay; certainly if we ignore gravity, then the ball would just keep going forever. But we want to keep gravity in there, but you're cluing into the right thing, which is what, the gravity on the moon is bigger or smaller than Earth; smaller. So if G is smaller, it's in the denominator, this whole number gets bigger; exactly right. X final will be much bigger. You could throw that baseball a lot further on the moon than you could on the Earth, okay? All right; any more questions about this approach? Yes. >> This might be off-topic, but what if G goes to zero? >> Ah, good. Well, that's sort of related to the question over here, right? What if G goes to zero? What does something over zero become? >> Infinity, I suppose. >> Well, let's think about it, right? What if G equaled zero? If G equaled zero, then we would get the following, X final equals VX initial 2H over zero. And what does that become? If I have numbers in there for H and VXI, and I divide by zero, what does that become? Yes; in the back. >> Infinity? >> Infinity; right, something over zero becomes infinity. And so it says that you would be able to throw that baseball infinitely far away. And that makes sense to you, right, if I threw this horizontally, and there was no gravity, this thing never curved down, it would just keep going forever, and ever, and ever, and ever, and ever off to infinity; to infinity and beyond. [Laughter] Can you tell I have young kids at home; yes, okay. All right, good. Hopefully, this is clear. Excellent questions. Any more questions about this one? Thought experiments? Yes. >> If gravity was zero and the direction of the ball was down -- >> Yes. >> So it would keep going in the same direction; it wouldn't change? >> That's right. If there's no gravity, if there's nothing to bend it, then it will keep going in a straight line. And we're going to learn in a little bit about Newton's Laws, and you've just identified one of them, Newton's First Laws, which is objects in motion tend to stay in motion. And the reason that they don't tend to stay in motion is because forces act on them, namely gravity. So if you are in the middle of outer-space, and there's no gravity anywhere, and you throw an object, it will go in a straight line forever, and ever, and ever. Now, that's not the universe that we live in. Okay; there's nowhere in the universe that you can escape gravity entirely, because there are lots of stars, and planets, and black holes, and all sorts of crazy neutron stars and things like that, and there is in fact gravity everywhere in the universe. So it's really a thought experiment, right, you cannot remove gravity entirely. But if you could, things would just keep moving in a straight line forever. Excellent. Okay; hopefully this was clear. If not, come see me in office hours. Cheers.