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 they're 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 it's 100 kilometers per hour. so i'd make it 100 units long. and let's suppose i'm 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 speed's gonna be what? this, take away this, that's gonna be that much shorter, sort of like that, you know? and if you're going-- again, and this time,

the wind is coming behind you, a tailwind. and you're 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? you're 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 don't be needing vectors. but i'll tell you where the vectors do come in handy. like, you got a crosswind, okay? let's suppose you're flying like this, and let's suppose the wind is a crosswind coming like this, just as fast as you're going. let's suppose you're going 100 kilometers per hour, and you're in a hurricane. and a hurricane is coming like this at 100 kilometers per hour. what's the direction of the aircraft gonna be? it's kind of easy to see, isn't it? it's kind of going like this. it's 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 that'll 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 parallelogram's 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? that's the direction that the aircraft will travel. and, furthermore, it tells you how fast it's gonna travel. because if this is 100, and this is 100, and that's 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 you'd be going something like 140 kilometers per hour that way. isn't that kind of neat?

let's suppose instead the wind weren't so strong. let's suppose you're 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 i'm gonna do in a minute, in a minute, i'm gonna go ahead and draw the solution. and you guys are gonna look. but why don't you beat me to it? why don't you draw what i'm gonna do next to show sort of this thing here, but the angle is different, yeah? why don't you do it before i do it? go. which is to say, you'll make a little rectangle, yeah? you see how now this is a rectangle? and what's the diagonal of that rectangle represent, gang? - the direction. - that's right. now, you'll 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 you're 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, we're really talking about direction too, yeah? and so we can represent that direction with an arrow. you know, i'd take this bowling ball and i roll it across the table, okay? as it rolls across, let's suppose i took strobe pictures: one here, one here, one here, one here, equally spaced times. you know what? i'd find equal space times would get equal spaces of distance. do you know why? because the ball is rolling at constant velocity. i mean, it's rolling here, then here, then here, then here. so my little velocity vector would simply be like this. i'm trying to draw that, so all these are the same, why?

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 it's rolling at constant velocity. let's suppose, instead, i take it and i drop it. [descending whistle] oh, what's 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 it's accelerating. and we've 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. what's the numerical value of g for the planet earth? check your neighbor. don't say 9.8, let's round it off. what's it gonna be, beginning with a t?

10, okay? all right, it's gonna be 10 meters per second every second, right? that would be acceleration due to gravity. and because it's accelerating, it's gonna keep going faster and faster, true? that's why i make the arrow a little bigger here. it's faster here than here. and over here, i'll make it like this even more. and that's because the speed, the speed picks up according to the acceleration and time. and the longer the time, the faster you're going, but you know that anyway. and so this is the relationship we've talked about last time. how fast something goes depends upon how much it's accelerating and how long a time it's doing that, isn't that true? let's 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, it's gonna pick up to 10, okay? let's suppose another second goes by, how fast is it gonna be going? 20? let's suppose 10 seconds go by.

it's 100, see. because every second goes by, it's gonna be going 10 meters per second faster than the second before. understand td t's talk about th. and let's suppose you're downtown and you're 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 couldn't see it then. check and see if you're 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, that's right. because every second go by is gonna pick up 10 meters per second more than it had before, we're learning the stuff, huh? let's 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. let's 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. what's 7.9 seconds, gang? how fast is it going? 79. 79, whatever it is, okay? it's 7.9 multiplied by 10, isn't that neat, huh? so that, well, then we review, okay? now, here's the more important question. lee, question. yeah, so let's 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. that's right. starts up at zero.

- as it starts going down... - that's right. ...i'm having trouble with that. that's right. here it's 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 we're saying is there's a relationship for how much it does pick up. it's simply g. and knowing how much it picks up, then tells you how fast it's going. and that's just g multiplied by t. is this too abstract, gang, when i say g and t and things like that, and i've got alphabet? you know, some people like numbers, okay? but this is--this 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 we're talking about, can't you? when you take music, hon, you gotta learn how to read the music, correct, yeah? isn't that right, nick, okay? well, what we're doing is we're learning how to read the music of physics. the relationships, and we're just putting them in a musical way, okay? how many people are tone deaf? okay, i am a little bit. but, anyway, that's what we're gonna do here.

does that answer your question then? so that's the speed we're gonna pick up. now, i'm gonna get to the part that's a little difficult for some people. that's how fast it goes. how far is it gonna fall? oh, we'll 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] it's 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, there's probably no rule for that, it's different every time. come on, gang, what's 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 let's suppose 10 seconds went by, you're in the airplane. you're dropping from the airplane. it takes 10 seconds to hit the ground. so if you're 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? it's 100, honey, 100, okay? so now you got 100 times-- it's gonna be 500 meters down. so if you're 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." that's how far you'd go. and you see that. any questions on all this? so we're really summarizing what we talked about last time. there's a difference between how much you pick up speed, okay, and between how fast you're going and how far you're going. and for falling things,

the accelerati same, 10 meters per second per second. later on we'll 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 you're falling. it makes sense. and how far you go has to do with the time 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 don't expect you to derive that. i want you to just know this doesn't pop out of the air and magic, okay? there's a reason for this. now, we talked how that came about last time. look it up if you want. what's kind of interesting is the behavior of the ball when it rolls off the table? i don't think i'll roll it off the bottom. it might wake up the people downstairs. but can you--it's gonna kind of curve, isn't it, gang?

it's gonna curve. and, you know, that curve wasn't understood for a long time. and when that curve was understood, it was kind of exciting. and let's talk about that. we can kind of show that up here. it's all on your book. and suppose i roll the ball, the ball is rolling at constant velocity. i got it going somehow, but once it's rolling, it's rolling. and let's suppose there's no gravity at all, none. and what's 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, it's gonna keep on moving unless something messes with it. and so that when that rolls off, if there's 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. push it, it gains speed. but if i don't push it, it will just keep rolling steady, steady, steady with no change. and, furthermore, there's 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 there's no gravity, would continue steady, steady, steady. but it doesn't 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 i'll 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 doesn't. it never does get out to there. it falls underneath, guess how far underneath it falls, one guess? all right, two guesses. [laughs] it's 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 what's 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 can't draw that down here, but i can see you get the idea. now look what happens, see the speed here, i'll 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 it's 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, it's like the airplane flying into the wind, the crosswind, huh? what i get is like that, it's 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 here--and now my vertical component longer and i get something like this. so the path of the ball really is-- that's 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 doesn't change. it's only the downward part that changes. and you know why that's 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] isn't that nice? take a cannon ball and you throw it up in the air like that, okay? here's the speed like this, the speed gets less, less, less,

now it's about to pick up. all along here, the sideways part stays the same. the sideways part doesn't change, only the vertical part does. and that's kind of neat. and human beings didn't know that for a long, long time. the sideways part doesn't 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? how--what kind--how 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.

iohat. 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 it's vertical, 'cause it will always have this component of speed, you know? it's like, where i live-- i live at a place on the 32nd floor of a building, okay? and down here there's 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, here's the thing that's kind of interest. you kind of look over there and you wonder if you could make it. well, it turns out i'm so high that it'd 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'll give you a hint, ss. same same. you not be knowing that? let's try it. and i'll come back to this swimming pool thing in a minute. there's a little device here that will shoot a projectile, a little spring, a little spring here. i'm gonna compress the spring.

and when i do that, i can put a projectile here, and watch this, gang, a little spring gun, okay? out it goes, no surprise. but i've got another ball that i can put on the end here like this, and when i go like this, it falls essentially straight down, a little bit out, okay, but kind of downish. this one goes outish. now, i'm gonna do them both at the same time. when i do them both at the same time, you guys got to figure out, hey, which one is gonna hit the ground first, the one that just drops down, or the one that... [whistles] ...goes out? a lot of people think, "oh, the one that's fired out is gonna be in the air for a longer time." do you know why? because gravity gonna start to pull it. "oh, i didn't know you're moving, go ahead." and gravity not gonna pull so hard. what do you guys think? let's try it. i tell you what, i tell you what, if you hear this, all a's. [laughter]

all--for this course, all a's. registrar would say, "you got all a's in your class?" i'll say, "i had a sharp class, honey, i had a sharp class, all right?" all a's. but if you hear this... [tap] then you got to do your homework and everything, okay? here we go again, here we go. this next moment may be very important for a lot of you who have everything riding on your grade. all right, here we go. here we go, listen. what did you hear? how many heard this? wishful thinking. we'll do it one more time in slow motion, all right? now you can get a better look this time. slow motion, here we go. all right? same time, gang?

yeah, gravity doesn't take a holiday on an object just because it's moving. it's gonna take the same time to come down. and so we find that over here too. drop one ball, throw another ball out, same time. since you have that--with things just dropping straight down, it goes 10 meters per second. but if it also is going horizontally, doesn't that object have to go-- be going-- traveling at a faster speed than the one that's just dropping vertical? if you're gonna get an angle that's more of this way, yeah. see what happens in freefall though that the vertical speed will keep increasing, increasing, increasing and become very, very big compared to that horizontal speed you have. so it looks like you're going straight down. what do they do in the old cartoons where the roadrunner runs off the cliff, you know? the roadrunner runs like this: da-da-da-da-da-da-da, aaaaah! foooom! okay? they go straight down. but that's a little bit different than in that, gang, isn't it? the roadrunner will be like, doo-ooo-ooo. goes like that. always going this way all the way down though. a lot of people don't realize that. they think you go like this and then straight.

you don't go straight. there's always a little angle. and that angle gets steeper and steeper as time goes by. but here's the point i want to make, gang. if it takes four seconds to go from here to here, it takes four seconds to go from here to here. so i ask people when they get up there, do you suppose if you jump, you could hit the pool? and how would you find out? and here's what you do. you figure out how fast, how far out you could jump in one second, okay, in one second, then multiply that four times and you got it. it turns out you'll really hit way down here. let me--i hit something like way in here, see? in one second, i can jump that far, two seconds that far, three seconds that far, four seconds that far. so it turns out, it gets very, very steep very, very quickly and you get so far. knowledge of physics. you people are gonna find out that as you learn more and more physics, your social life is gonna increase

because people are gonna gather around you. people are gonna find you more interesting, okay, because you're gonna have things to talk about now, okay? just like the time years ago, a fellow came to the town in san francisco and was kind of lonely. he didn't know anybody. he wanted to make some friends, so what did he do? go to church on a sunday morning, huh? he went to the church on a sunday morning, the good reverend says, "hey gang, we're gonna meet "this afternoon at 2:00. we're gonna take a bus trip. "we're gonna take a bus trip out to hang-gliding territory "out on the cliffs over the beach. so all the people get together and they get in this bus and they take--go way out where the hang gliders are in these cliffs, okay? and, like, people got out of the bus, they spread out the big cloth and put the kentucky fried chicken, the kool-aid, and all the stuff there, the mcdonald's and all the other things right there, right? this guy is sitting down and kinda not checking out the food, checking out the potential friends, looking around, he's trying to find out who's who. he looks across the way and sees someone and said, "whoo, this is it." boom-boom boom-boom. is it? okay, finally happened. and he looks at this lovely, lovely person.

a lovely person who walks over to the edge of the cliff and looks down and says, "gee, i wonder how high we are above the water?" how high above the water, do you guys remember this? how high above the water, i said, "hey, i can do that." some other guy gets up, walks over to the edge and say, "how high we are above the water, "here watch this. watch this rock. [descending whistle] "five, five seconds. "5 times 5 is 25, 25 times 5-- we're 125 meters high." delightful person looks at that guy and says, "wow, okay." and they're getting together fine, you're sitting over here and what are you doing? [pretends to cry] "i could have done that." you miss out. you miss out. you got to be fast, you got to be fast. lonesome city, honey, lonesome city. you strike out.

they're getting along . at least you can feel happy for them, right? next week comes, you go to another church, the reverend type says, "all right, all you lonesome hearts "out there, we're gonna get together today, "1:00 today we're gonna get a bus trip together, "we're gonna go picnicking. "so all you new folks, come on, we all get to celebrate life. we get to meet each other, come on, 1:00 come on, huh," 1:00 come, you get in a bus. different color bus, this time it's a green bus, okay? you get on the bus, you go out to the hang-gliding territory, watch the hang gliders soar, huh? a little further upstream, okay? you're sitting down, they put out the big tablecloth, the kentucky fried chicken, the kool-aid, the mcdonald's, the burger kings, whatever. all of them right out there, right? you ain't checking out the food. what are you checking out? potential friend again. and son of a gun, there's an even more delightful person, okay, bam, bam, bam, bam. gets up, walks over to the edge and says, "i wonder how high we are above the water?" you're back here and you say, "santa claus has come to town."

you get up, you walk over, you grab a rock, you say, "watch this. i'll tell you how high we are." you take the rock and you drop it, all right? you drop the rock but you're a little bit further upstream now, right? a little different place, okay? and you dropped the rock. [laughter] and you count off the seconds, you say, "according to my calculation, we're several kilometers high." what's this delightful person say to you? "you schmuck, i'm not gonna go home with you, okay?" you--"you don't have to do that." how could you do that? how could you impress that delightful person? there's a way to do it. okay, we haven't talked about torques yet. but if you knew about torques, you could do it. let me show you what to do. what to do is you don't drop the rock like that. you get your board, long plank. then you roll a great big piece of rock on it like that.