We start out by talking about vectors. The vectors. Do you know what a vector is . Like an arrow. If you ever sit next to a physics type and theyre doodling, giving ideas, talking about things. Usually, they put a whole lot of arrows to represent things. An arrow is a vector which represents, like, which way and how much, you know . Like, when we talk about flying into the wind. Notice in the textbook, we talk about the airplane flying into the wind. And we can let the speed of the airplane maybe the airplane is going due north, Something Like this, okay . We could let that arrow be the speed. Maybe its 100 Kilometers Per Hour. So id make it 100 units long. And lets suppose im going into a headwind, and the wind is coming toward me at 20. Well, i can go like this, see, make an arrow 20. When i combine these things, my ground speeds gonna be what . This, take away this, thats gonna be that much shorter, sort of like that, you know . And if youre going again, and this time, the wind is coming behind you, a tailwind. And youre going, again, 100 with respect to this still area. And then if you got say, a 20 coming behind you, those two things combine to be what . Youre going faster. Do you need vectors to explain this idea, that if the wind is behind you, you go faster, and if the wind is in front of you, you go slower . Answer begins with the n. No, you dont be needing vectors. But ill tell you where the vectors do come in handy. Like, you got a crosswind, okay . Lets suppose youre flying like this, and lets suppose the wind is a crosswind coming like this, just as fast as youre going. Lets suppose youre going 100 Kilometers Per Hour, and youre in a hurricane. And a hurricane is coming like this at 100 Kilometers Per Hour. Whats the direction of the aircraft gonna be . Its kind of easy to see, isnt it . Its kind of going like this. Its kind of going like this at the same time, right . And so what it does, it kind of goes off course like this, huh . Let me give you a neat little rule thatll tell you exactly which way and how fast it goes. And the little rule is this. Take your two vectors, one representing ground speed or the speed through the air, and the other representing the speed of the wind, and make those into a parallelogram. Since they are right angles here, that parallelograms gonna be, in this case, a square, because the sides are equal. Make a square. And then what you do is you join from here to the diagonal, and you make a vector like that. And guess what, gang . Guess what . Thats the direction that the aircraft will travel. And, furthermore, it tells you how fast its gonna travel. Because if this is 100, and this is 100, and thats a 45 degree angle, and it would be for both 100, yeah . It turns out, so this will be this will be the square root of 2 times 100, Something Like 140. So youd be going Something Like 140 Kilometers Per Hour that way. Isnt that kind of neat . Lets suppose instead the wind werent so strong. Lets suppose youre traveling like this, and you got a crosswind about like that. Now, at your seats there you guys taking notes, yeah . At your seats, draw this. And what im gonna do in a minute, in a minute, im gonna go ahead and draw the solution. And you guys are gonna look. But why dont you beat me to it . Why dont you draw what im gonna do next to show sort of this thing here, but the angle is different, yeah . Why dont you do it before i do it . Go. Which is to say, youll make a little rectangle, yeah . You see how now this is a rectangle . And whats the diagonal of that rectangle represent, gang . The direction. Thats right. Now, youll be blown off course only that much. And you know what, if you had a ruler and you did this to scale, you could measure that compared to this, and you could tell how fast youre gonna go, because the velocities will add together in that vector way, really kind of neat. Any questions about that . So when we talk about how fast things go and we talk about velocity, were really talking about direction too, yeah . And so we can represent that direction with an arrow. You know, id take this bowling ball and i roll it across the table, okay . As it rolls across, lets suppose i took strobe pictures one here, one here, one here, one here, equally spaced times. You know what . Id find equal space times would get equal spaces of distance. Do you know why . Because the ball is rolling at constant velocity. I mean, its rolling here, then here, then here, then here. So my little velocity vector would simply be like this. Im trying to draw that, so all these are the same, why . We got the same speed along. It goes at uniform velocity. It takes a little push to get it going. But once i get it going, i let go. It kind of rolls over on its own, and it rolls steady, steady, steady, steady, okay . And so these little vectors would just show me that its rolling at constant velocity. Lets suppose, instead, i take it and i drop it. [descending whistle] oh, whats it gonna be now . Here is it up here, see, then i drop it and it goes to here, and to here, and to here, and maybe down in here. Now up here, no velocity. But over here, a little bit. And over here, a little bit more. And you know the velocity is increasing, because its accelerating. And weve talked about acceleration. And when the acceleration is due to gravity, we just call it g, remember that . And we know that g for the planet earth is check your neighbor. Whats the numerical value of g for the planet earth . Check your neighbor. Dont say 9. 8, lets round it off. Whats it gonna be, beginning with a t . 10, okay . All right, its gonna be 10 meters per second every second, right . That would be acceleration due to gravity. And because its accelerating, its gonna keep going faster and faster, true . Thats why i make the arrow a little bigger here. Its faster here than here. And over here, ill make it like this even more. And thats because the speed, the speed picks up according to the acceleration and time. And the longer the time, the faster youre going, but you know that anyway. And so this is the relationship weve talked about last time. How fast something goes depends upon how much its accelerating and how long a time its doing that, isnt that true . Lets do a review. If i took this thing and held it up in the air and let it go. [descending whistle] at the end of one second, what would its speed be . 10 meters per second, see . Because it started at zero, its gonna pick up to 10, okay . Lets suppose another second goes by, how fast is it gonna be going . 20 . Lets suppose 10 seconds go by. Its 100, see. Because every second goes by, its gonna be going 10 meters per second faster than the second before. Be sure you understand that, and lets talk about this. And lets suppose youre downtown and youre looking out the window, and all of the sudden, this thing falls right by the window. You know, according to my calculations, that bowling ball just fell by the window at 50 meters per second. Someone would say, how did you know that . Well, i just know that. One second later, boom, hits the ground. Okay, how fast was it going when it hit . You couldnt see it then. Check and see if youre sitting next to somebody who know the answer to a question like that. How many say 60 meters per second, show a hands . Hey, my people, all right, all right, thats right. Because every second go by is gonna pick up 10 meters per second more than it had before, were learning the stuff, huh . Lets suppose i said one second after that, what would it be . 70. 70. So it keeps picking up the same well, here it is right here. G is gonna be 10. Lets just call it 10. If g is 10, just multiply by the number of seconds. Even if i told you it falls in 7 seconds, you can still do it. Maybe not 9 seconds. Okay, 9s are kind of tough. But see, 7, all right . 7, you just say 7 times 10 is 70. How about i gave a decimal 7. 9 seconds . Oh, remember the first time you saw a 7. 9, oh, god. Whats 7. 9 seconds, gang . How fast is it going . 79. 79, whatever it is, okay . Its 7. 9 multiplied by 10, isnt that neat, huh . So that, well, then we review, okay . Now, heres the more important question. Lee, question. Yeah, so lets see, last time we were talking about throwing a ball up in the air and then having it drop. Yeah. And so at the top, its velocity will be zero. Or when we originally let it go, its velocity is zero. Thats right. Starts up at zero. As it starts going down. Thats right. Im having trouble with that. Thats right. Here its zero, right here, lee, okay . I drop it. Now it picks up, picks up, picks up, picks up, da, da, da, da, da, and all were saying is theres a relationship for how much it does pick up. Its simply g. And knowing how much it picks up, then tells you how fast its going. And thats just g multiplied by t. Is this too abstract, gang, when i say g and t and things like that, and ive got alphabet . You know, some people like numbers, okay . But this isthis is alphabet. Remember when you first get into that, you get into algebra . You looked in the algebra book and there are no more numbers, all alphabet, alphabet equations. What if this is . Does this turn anyone off . You can kind of see were talking about, cant you . When you take music, hon, you gotta learn how to read the music, correct, yeah . Isnt that right, nick, okay . Well, what were doing is were learning how to read the music of physics. The relationships, and were just putting them in a musical way, okay . How many people are tone deaf . Okay, i am a little bit. But, anyway, thats what were gonna do here. Does that answer your question then . So thats the speed were gonna pick up. Now, im gonna get to the part thats a little difficult for some people. Thats how fast it goes. How far is it gonna fall . Oh, well get it all get mixed up. How fast, how far . How far is different than how fast, right . And how about it, gang, when i drop this thing . [descending whistle] its gonna pick up distance, yeah . You see it getting further, further apart, huh . What is the rule for how far it falls . Is there a rule . How many would say . No, theres probably no rule for that, its different every time. Come on, gang, whats the rule . Do you remember . Yeah, it was distance falling, d for distance, equals, average out the g, g squared. And if g is gonna be 10, and it will be for the planet earth, then a half of 10 is 5, so we could just say, 5t squared. So we should be able to find from here that that distance keeps getting greater and greater and greater, greater for time. And that being true, you can answer this question. I take this falling ball, i get up on top of a cliff and i drop it. [descending whistle] how far down is it underneath one second later . Check the neighbor. How many say begins with an f, ends with a ive . [laughter] yes, five meters, five meters down, okay . Remember that any object that falls from rest will fall a vertical distance of five meters in the first second of fall. If two seconds goes by, how far will it have fallen then . Even more. How much total and how would you find it for two seconds, gang . You got your little rule right here. If two seconds goes by, you take 2 times 2 is 4 times 5, 20 meters. It would have fallen 20 meters. And lets suppose 10 seconds went by, youre in the airplane. Youre dropping from the airplane. It takes 10 seconds to hit the ground. So if youre sitting next to someone who knows how fast i mean, not how fast, how far down the ground is . Well, 10 plus 10 plus 10 plus 10 plus 10 plus 10 plus 10 plus 10 plus 10 or what . Its 100, honey, 100, okay . So now you got 100 times its gonna be 500 meters down. So if youre ever falling off a cliff and it takes 10 seconds to hit the ground, your last thought will be, hey, it took i bet you i fell 500 meters. Thats how far youd go. And you see that. Any questions on all this . So were really summarizing what we talked about last time. Theres a difference between how much you pick up speed, okay, and between how fast youre going and how far youre going. And for falling things, the acceleration will always be the same, 10 meters per second per second. Later on well round that off to 9. 8 meters per second, per second, okay . But 10s are easier to deal with, right . And we find out that how fast we go is simply the acceleration multiplied to how long youre falling. It makes sense. And how far you go has to do with the time squared. So its averaging out 5t squared. How we got that is derived in the footnote of your book, and you can kind of go back and look at that if you want. I dont expect you to derive that. I want you to just know this doesnt pop out of the air and magic, okay . Theres a reason for this. Now, we talked how that came about last time. Look it up if you want. Whats kind of interesting is the behavior of the ball when it rolls off the table . I dont think ill roll it off the bottom. It might wake up the people downstairs. But can youits gonna kind of curve, isnt it, gang . Its gonna curve. And, you know, that curve wasnt understood for a long time. And when that curve was understood, it was kind of exciting. And lets talk about that. We can kind of show that up here. Its all on your book. And suppose i roll the ball, the ball is rolling at constant velocity. I got it going somehow, but once its rolling, its rolling. And lets suppose theres no gravity at all, none. And whats gonna happen . The ball is gonna just keep rolling like that. And if i have an arrow representing the speed at every time, it might be Something Like this. And you know why that speed stays the same . It has to do with the idea we call inertia. When something is moving, its gonna keep on moving unless something messes with it. And so that when that rolls off, if theres nothing pushing it this way, then it would just keep going at the same speed. Like when this ball is rolling along and now i push it. When i push it, it gains speed. But if i dont push it, it will just keep rolling steady, steady, steady with no change. And, furthermore, theres nothing this way obstructing it. Now, there is some air drag, but very, very little compared to the tendency of that ball to just keep crashing through. So the ball goes steady, steady, steady. And rolling off the tabletop, if theres no gravity, would continue steady, steady, steady. But it doesnt continue like that, because there is gravity. And gravity pulls it down. What gravity is gonna do is like this. The ball we just hear it dropping, if i let it go, it falls here maybe. Then falls in here, then falls in here, and then maybe fall down to here, okay . And then ill have like a little speed like this, a little more like this and a little more like that, and down here even more. What happens, as the ball is traveling like this, it does this. It falls. So when it gets out to here, it really doesnt. It never does get out to there. It falls underneath, guess how far underneath it falls, one guess . All right, two guesses. [laughs] its gonna fall right to here. And instead of getting out to here, gravity would have pulled it down to guess how far . It will exactly match whats happening over here. Kind of neat, huh . It will fall to here. And then instead of getting out to here, and no table to support it, it will fall underneath. Guess how far underneath such as to match right here. And instead of getting to here, it will fall on down to, i cant draw that down here, but i can see you get the idea. Now look what happens, see the speed here, ill put that over here, put this one here. And when i combine them, i get a speed like that. Can you see that . And when its over here, it still has that speed sideways, because nothing is messing with it sideways. That part remains. But now gravity has pulled it down. So it has the same speed it would have over here. So i put this over here. And when i combine them, its like the airplane flying into the wind, the crosswind, huh . What i get is like that, its picked up speed due to gravity. And over here, it has the same speed here. We call this the horizontal component of the speed. We call this the vertical part or the vertical component. Component is more fancy than part, huh . So we have a horizontal component and a vertical component for every one of these speeds. Over hereand now my vertical component longer and i get Something Like this. So the path of the ball really is thats not drawn very well now. It should be a little more like that. And, hence, the ball curves. But the neat thing is that what happens sideways doesnt change. Its only the downward part that changes. And you know why thats true . The gravity pulls which way again . Beginning with a d, ends with a own. Try it. Down. Down, gravity pulls down. And so guess which way it accelerates . Down. Down. How about accelerating sideways . No. So the sideways part stays the same. [mwah] isnt that nice . Take a cannon ball and you throw it up in the air like that, okay . Heres the speed like this, the speed gets less, less, less, now its about to pick up. All along here, the sideways part stays the same. The sideways part doesnt change, only the vertical part does. And thats kind of neat. And human beings didnt know that for a long, long time. The sideways part doesnt change. Question . And one thing that might have, maybe occurred without figuring it out, is the question of how long does it have one of those speeds drawn in there . Howwhat kindhow long does it have that speed . Are any of those an average speed . Yeah. Well, no, this would be the speed at any instant, lee. Now, in the absence of air drag. It turns out with a projectile usually there is a lot of air drag like with a cannonball. But in the absence of air drag, that horizontal component of speed stays the same. That means it will always be kind of going this way. Ill tell you an interesting application of that. You know, when you throw a rock off a cliff, it keeps going out, out, out, and it gets steeper, steeper, steeper, right . It will never get so steep that its vertical, cause it will always have this component of speed, you know . Its like, where i live i live at a place on the 32nd floor of a building, okay . And down here theres a Swimming Pool, and up on the top sometimes i think about, well, rather than getting the elevator, come all the way down, and have to come out open the door, go through the gate, come in and why not just. [descending whistle] and i have to wonder if i could make it [laughter] now, heres the thing thats kind of interest. You kind of look over there and you wonder if you could make it. Well, it turns out im so high that itd take about four seconds for an object to fall, okay . If i just stepped off the balcony, it would take four seconds to hit the ground. Let me ask you folks a question, how about if i jump out sideways. How many seconds . Beginning with a f, and with a our. Try it. Four. Same four seconds. You see what i mean . This ball that rolls off the table, takes the same time to hit, as one that drops. Do you believe that . Remember the first day we talked about the rifle . You take the rifle and you fire it, and you let go of the bullet. And one bullet falls down, and the other goes out, which one hits the ground first . We talked about this. Which one does hit the ground first, gang . Take a guess. I