 All right, any questions, comments, concerns, Phil? Oh, I hope you have the lab from yesterday here. That's not due next Monday. No, no, it's going to be part, you're going to combine it with the work from last week as we start building up this bigger report. So there's no lab due? No, not exactly. But I sure hope you've got it here. Actually, you don't need to because I have the very same data so I've worked it up so we're going to be able to talk about it here. But if you've got it, break it out because that's what we're going to look at today. So far we spent the first part of the term looking at what we call rectilinear motion. That was the one-dimensional motion we started with. That's where we introduced the concept of position and velocity came from that. We looked at both average and instantaneous velocities. And then we took the step to acceleration. We looked at both average and instantaneous acceleration. Now we're looking at curvilinear motion. And we started sort of a touch on that on Monday when we started looking at vectors because that's a full two-dimensional representation of what's going on. The vector's very, very good for any parts of that we need. Whether it's the position, just simply as x, y coordinates that helps us have then the length of the vector and its direction. And of course, whatever the length is, there's got to be units with it. But then we also do the very same type of thing that we did with it in terms of velocity and acceleration. So we introduced this idea of a position vector on Monday. And we looked at it, there's different ways you can represent it. The main way we did it on Monday was we just simply talked about the two orthogonal components, the two vectors, one horizontal, one vertical, that we could add together and have the very, very same thing. And this is very much like you would have done for a little trip across town to your friend's house. You wouldn't cut straight across all the yards. You'd first go somewhere down Elm and then up on Main. And that's how you get there. That's exactly what we mean in these two instances. This is simply an analytical representation of this very simple idea. The reason that's useful to us is as we go to add vectors together, it's much easier for us if we add them together. It's much more accurate if we add them together by adding up all the r parts of all the vectors we might have to make up the x part of the, whatever the resultant might be. So if we've got a couple vectors in here we're going to add together. And we did this very thing on Monday with one of the homework problems only. I think I called the vectors a, b, and c at that time. But we can call them r1, r2, and there could be however many there might be. Fun to know what the resultant of all those vectors being added together would be. Don't forget they could be subtracted too. Subtraction is just sort of an addition backwards. And so we added up all those little vectors. Is everybody familiar with this summation sign notation? It's going to be real important to us. We are doing more with it than just that. It's just our symbol that says add up all these things that we've got in this problem. We're talking about all these different position vectors we might be adding up. Whatever they all might do. We had a problem on Monday that had, that had three vectors. Then one of the homework problems that follows had a few more vectors in it. Doesn't matter. However many there we can add them all up. And the easy part of it is for us, especially since we all know trigonometry, is we can take any vector, break it into its orthogonal components, add up the components themselves, and that will give us the resultant vector. The only thing I'm going to add here is an extra, a slightly different notation that we might use. I can write it as I've shown here. We also have another way to write vectors. Remember all vectors have magnitude in units. That'd be whatever number appears here as the size, the distance in the x or the y direction we're going. Also though all vectors have directions. So one way we can write this is just whatever the magnitude is. Notice the vector sign's not on the top anymore. But you're all probably screaming inside because this offends us. Aren't you feeling it? Who's feeling offended out there? Joey is. Yeah. Len can't stand. Phil's just dying to come up here and fix things. What's wrong? What's wrong is I said a vector equals something that isn't a vector. There's no direction in here other than the subscripts, but if I just end up putting some numbers in there like three miles and six miles, where's the direction part of that? So I need some direction to complete this as a full vector representation. So what we do is the simple idea of what we call unit vectors. They're extremely useful to us as a notational key. It's the only vector you'll ever see that has direction, magnitude, and no units. Its magnitude is one. One nothing. Just one. A magnitude of one. It's kind of like the sine and the cosine and the tangent and all that stuff, that those never have units on them. Those are just distances but in comparison to a unit circle, if you remember how you first learned trigonometry, but there was never units on them. No cosine that you ever take will have units on it itself. And they give us direction in a particular way. We're going to have three unit vectors we use in this class. The first will be a unit vector in the x direction. And we call it I-hat. So I'll put that just after that Rx thing. Now I'll have, if I had numbers in here, I'd have magnitude, units, now I have direction. This is x direction. Sorry? I-hat is the I unit vector or the x unit vector. Or we say I-hat. Don't we say that, doesn't that sound okay? See a little hat on it? And you can do this in math type when you buy it for a buck and a half. We also need one for the j or the x direction. That's called j-hat. So I'll put that on there and we've got everything we need. Do me a favor. I noticed the students new to this are a little bit sloppy. Make your i look something different than your j. Look down on your paper right now. Make your j look a little bit different from each other. That way I'll recognize them that way. You'll recognize them. Either one of these notations is fine. It's just that this one, once we take out the R sub x and put in a number and units that go with it, then the direction isn't so clear anymore. In this version, the x is what gives us the direction. The y is what gives us the direction. But if we put values in there, where are the x and the y? We don't put an x and a y. I wouldn't write three miles sub x. You might try it, but I wouldn't do it. So we can always put something like, for example, three miles in the i direction plus six miles in the j. And now you know exactly what to do. Get your little buddy's house. You're going three miles in that direction and then six miles in that direction and you knock on the front door. And there for you is milk and cookies. Yeah, Julie likes that. That sounds good. Right about now. Maybe some warm milk. Be nice. If anybody all of a sudden feels you're, it's called mom. Go ahead. So we're going to use that sum. It's also good for us because it helps us when we do these summations over here, we'll only sum out values in the i direction together. We'll only sum values in the j direction together. When we go to add a vectors, we're not going to mix and match those. Yeah, they're added in this way, the two components being added together. What we're going to do is keep all this stuff separate and it just becomes a bookkeeping problem for us then. As we go to add up all the vectors, we add up all the x ones together, add up all the y ones together, give them each their proper direction and we're all done with the problem just like we showed on Monday. Takes a little practice to get used to. But that's what the homework's for. That's what my full homework solutions are for to help show you how we go through this stuff, how you get used to it. For most of you, I assume this i and j, in fact, we'd have an x direction if we did a three-dimensional problem. Then we have a k hat. For most of you, is that notation new? I would assume so. But again, it's just a bookkeeping aid for us so that we understand when I'm adding certain magnitudes together, I add the x magnitudes together and the y magnitudes. Remember, I'm not adding three miles to six miles. I'm adding three miles in that direction to six miles in a different direction and that's a completely different answer than that. I don't get nine miles. I get whatever that diagonal across there is having gone three and then six. So it's going to take some practice but we're going to work through it. Any questions before I clean the board and we get to what we discovered yesterday in lab? No, you're okay, you're comfy? Nothing really. It's just that when I take out the rx and put in three miles, now we need something that says that was three miles in the x direction because I'm not going to write three miles sub x but if I write three miles in the i direction you know exactly what I meant. I meant three miles in the x direction. In this form as they show, there's no difference but once the values come in then these little subscripts wouldn't be there and we need something else that represents direction on everything. It depends on the problem and remember how you set up an arbitrary reference, an arbitrary origin. If I have one vector that goes that way for three miles and then I have another one that comes this way for three miles I would write this one as three miles i hat this one is minus three miles i hat and that shows that they're the same magnitude measured with the same units both in the x direction but one's positive, one's negative they're in the opposite direction. Why is that direction negative and this one isn't? Who can answer that? Yeah, because I chose the origin. I just, we typically say that way's positive this way's negative so it's nice to stick with stuff that's typical if possible. Two good questions. Any others? So let's look at what we have in lab. Yes, everybody quite got to it let's see, let's... Oh, I want you to do this as we talk about this remember we're talking about projectile motion it's a two-dimensional type of motion obviously there's some travel horizontally and there's some travel vertically going on here so it fits really well into the types of two-dimensional motions we're looking at but I want you to do this with it as we start to look through what we did yesterday put down the middle of your paper and you can turn your paper sideways if you want a little more room on either side put down the middle... No, no, don't do it yet I didn't tell you what to put down the middle of your paper because just a simple line will not do it it's got to be a semi-permeable membrane don't put a line down the middle of your paper you've got to put down a semi-permeable membrane go ahead and do that now semi-permeable membrane, do it do it, keep going you've got to go down to the bottom so there's no leakage we don't want stuff getting through here that isn't okay to get through here you get it? where's your semi-permeable membrane? I want you to draw a semi-permeable membrane a lot of you are just going to put a line down the paper Joe, weren't you going to do that? but now you've got a semi-permeable membrane down the paper that's better now we can get some work done hey dude, good job didn't you? you can go to a new sheet of paper if you want I just said turn it sideways just so you have a little bit of extra room if we need it we looked at that video real quick of the guy throwing a projectile across the screen does that not be as awful? did as good a projectile motion as I possibly as we could have done in the lab so that's why it's a lot of you are just to use their video that video there picked out of it the position at certain times of that projectile as it went across the screen got a real nice parabola out of it expected it looked something like that most of you did catch that some of these values in here throw out the ball was still in the guy's hand projectile motion is what we call free fall motion here's what the definition of free fall is I gave you a pretty clear definition yesterday of what free fall motion is it's what we did with the tape drop experiment a week ago and then with the projectile motion yesterday the only force is gravity yeah there's air resistance but not a lot we didn't find a huge effect of it and I don't know if we want to actually run an experiment where there's no air resistance what it means is I have to pump all the air out of the room last time I did that the administration was very angry all the dead students I mean I knew what was going on so I had my school good tank on well they didn't call it peculiar they just said don't do it again is our only force that don't forget is a directional thing gravity only occurs why does gravity only occur downward an arbitrary question not rhetorical either that's a straight question I want an answer why does gravity only occur downward exactly that's where the center of the earth is it's the earth it's the object and the earth and gravitational attraction with each other if the object's up here and the earth is down there that's the only place that's the only direction gravity can act can only act straight down that's our first big clue to the nature of projectile motion the reason things do this now just a point in experimental work you cannot throw out points that just look bad and you don't like how they look these weren't just that they did work prior but we knew why we knew that when the ball was in the guy's hand there was more than gravity acting on that ball his hand was there in fact it was getting the thing moving and it was moving it in a bit of an arc and all those kind of things are other than gravitational motion so we knew that those points did not belong as part of the projectile and that's why we can reject them if you do an experiment and points show up that you don't like but you don't know why they don't look right you can't just throw them out you have to leave them in maybe you discover something no one else has ever seen before if you throw them out you won't get the Nobel Prize so you gotta leave them in you gotta think about them but you can't just throw them out because they're ugly other things in life you can throw out because they're ugly but no you can stay but not that once okay I throw those out just because they're ugly so that's really important to us because that's our first clue of what's going on here and why this parabolic path through space naturally occurred as we threw the ball so now that we've got this semi-permeable membrane down the middle of the paper we wanna decide what's going on on each side why have we done that what we're gonna do is separate the motion of what's going on of the horizontal motion here and the vertical motion there gonna do the other way around it doesn't matter what's on the left or what's on the right it does matter that they're separated by this semi-permeable membrane so we've already established that gravity is the only thing going on here once it's out of the guy's hand gravity is the only thing going on here and it acts in the vertical direction only so we know over here the acceleration in the y direction now remind me I think I did remind me if I gave you this notation y double dot did I give you that notation as an alternative evidently I did to Samantha and no one else did I? No? just a shorthand notation any time we take the time derivative of something which in this case is the y so we took that the time derivative of y I don't want to always write d dt so I write y dot just simply a shorthand notation so this is well if we take the time derivative of the y position what do we have when we take the time rate of change of the y position well we have the y velocity specifically so it's just a little shorthand notation for us what happens when we take the time rate of change of the velocity we get the acceleration so if I take d of the in the y direction dt y dot because they were the same thing well y dot is already a dt so this is d dt of what's already d y dt is that right? you took second derivatives in calc one this is just the second derivative you used it to find out concavity and all that kind of stuff well let's see I don't want to write all of that so we also shorten it down you've seen that notation from calculus in calculus a lot of times you write f double prime same thing it's just f it doesn't help us a lot we need to know exactly what it is for taking the derivative of and I'm too lazy to write that so that's all the double dot means it's the second derivative with respect to time so y double dot is the acceleration in the y direction and we happen to know that that's about minus g y minus say what? that's the way I chose it to be in this case in the down direction so it does kind of make sense to pick one or the other as a plus or minus when we did the tape drop about half of you took down as negative and half of you took down as positive didn't matter here we do need to be a little bit more careful so I'm going to decree because I'm a high master here that down is negative that's well for my aging brain we want to accommodate that so I say free fall you know that means gravity is the only force that gravity is the only acceleration we already know this to be true in projectile motion gravity is the only force it acts straight down what then is the acceleration in the horizontal direction is a sub x or x double dot if you'd rather I don't care which one you use it's up to you I recognize both you should recognize both here's the data from yesterday everybody got something something a little bit like that pretty much like that pretty familiar I hope notice that for the convenience of the reader right now I put that up for your benefit I adjusted it so the origin on both is right at the first dot cut off a couple dots change the origin so it's right there for the convenience of the reader which is you if it starts at some other time and then you have to think oh did I show up late for the party or if it starts at some other height you have to wonder the height of what, what was that it's immaterial to the reader because what's important is the parabolic motion the projectile motion itself so I think at least half of you we looked at how to use excel to very easily adjust that kind of thing again emphasizing that the origin the location of the origin is arbitrary it's not going to affect the rest of the physics that's going on alright so let's let's see, let's see what we have here we've got oh oh remember this remember we have the constant acceleration equations remember those, did I give you that sheet pull it out if you got it now the one that would go with this top graph actually it would go with either graph is the third one do it in slightly altered form the third one is delta y remember that's y2 minus y1 so I'm just going to break the y1 or it's yf minus yi I'm just going to break yi over to the other side yf equals 1 half y double dot I'm taking out the a and putting in the y double dot c squared does this look like the third equation so far is it the third one there it has s I'm putting in y because I'm talking about a particular direction and I've taken out the delta y broken it into yf and I'm going to have yi over here so let's see y dot initial t plus yi isn't that the third one cast in a particular form illustrating the vertical motion in the y direction is that that third equation slightly different form just slightly though because on that sheet it's supposed to be for any constant acceleration problem I'm talking about a vertical constant acceleration problem we all recognize this is supposed to be let's see this is 1 half a so when I retighted it I accidentally took out the 1 half but that should give us most of you had does anybody have their data from yesterday here with them did you get something like these numbers very close okay okay a little bit different you may have had an extra point in there or something at all that that was alright so we've got 1 half we'll use my numbers up there for all to see so 1 half times minus 9.11 well there's our estimate of y double dot the acceleration in the x direction plus what else we've got in there 4.22 t that must be y dot i what in the world is that yeah initial velocity velocity because it's dot in the y direction y dot that's velocity i initial velocity in the y direction let's see that must mean right at this point the initial point of the launch the first point we had where it was out of his hand remember we can't count any of the time it was in his hand there must be some initial velocity in the y direction oh I guess that makes sense that's why it went up because he threw it up a little bit makes sense why else would have gone up if he didn't throw it up a little bit Monday morning after the Super Bowl I'll tell you what I was thinking about not this class it was my other class I had a couple students that didn't show Monday morning plus what's the other part there a minus point o and that must be what's yi initial position that's pretty darn close to zero why isn't it zero the uncertainties in the measurement remember I was picking out those data points as the ball went across the screen did I hit exactly the same point on the ball every single time no of course not of course not but boy that's pretty darn small what is that that's a that's about three centimeters that's a difference about that and the guy threw it all the way across the room so that's not a big deal we won't worry about that little piece it's so close to zero but notice we did pick up two pieces of information the initial velocity in the y direction and the acceleration in the y direction and it's about what we expected it to be 9.8 Samantha's was actually a little bit bigger than mine was but it's certainly in the ballpark not quite as high probably because there's some air resistance going on it's messing some things up plus the uncertainties in measurement let's look not at the vertical motion let's look at the horizontal motion and that's what that looked like doesn't make sense to put a parabola through that it's pretty darn linear yours was too wasn't it once you got rid of those first little bits of points that were when it was in the guy's hand once it was freeing in the air ready to go that's pretty much what it looked like a constant acceleration situation to you what do you look for to tell if the acceleration is constant hang on this is the position versus time so what's the derivative of that slope give you what's the derivative what is the slope of a position time graph give you give you velocity is this velocity velocity constant how do you know it's a nice straight line the slope is pretty much the same everywhere if the slope is the velocity and the slope is the same everywhere the velocity is the same everywhere so we can say x dot is constant because the slope of our position graph for the x direction looks constant it's not force feeding anything with that am I Tyler are you okay with that comfy sorry oh yeah that's right cause I talked what happened with those is I accidentally left off the first decimal place I know this now so this was really 0 work a minute ago this is really supposed to be 0.5 and it rounded it off this is 1.0 this is 1.5 that's 2.0 2.5 I think several times you've heard my frustration with using excel or any other microsoft product as we've been going along here just a few weeks we've been together this is again one of those frustrations oh yeah that's a lot more clear now beautiful that's what the problem was though thanks for catching that it was just simply the fact that it was rounding off alright so everybody comfortable that with that position time graph we're looking at constant velocity because the position graph is linear position graph is linear the slope is the same everywhere the slope is the velocity the velocity is the same everywhere if the velocity is constant what's the acceleration 0 well that's why there's no 1.5 at squared value in there did anybody the parabola through this what was your A wasn't it very very small it's been very very close to 0 and I probably came along and said don't do a parabola through that it's linear data so we know that the velocity is constant acceleration to 0 however is that still a constant acceleration problem yeah it just happens we know what the constant is it's 0 0 is a constant so the constant acceleration equations will still hold so I'll write down the very same one we had like there x final equals 1.5 x double dot t squared plus x dot i t x i that's the very same constant acceleration equation in horizontal form look at all the y's are gone there's only x's left only x's should start to give you an idea of why I wanted this barrier down the middle of the page there are no y's welcome over here there are no x's welcome over here these are two gated communities where the horizontal people do not let the vertical people come to visit they're not welcome over here the vertical people do not like the horizontal people they're not welcome to visit over here this is a segregated society and will remain so no y's over here no x's over there welcome they can't make it through this barrier let's look back at this one our acceleration is zero because we have a nice straight line so that's gone we didn't even use it just by putting a linear fit in there I was already prepared to say it was zero so what did we get for a number .29 I'll just make it t minus 0.017 2.91 what let's look up here x dot means velocity it's the initial velocity in the x direction well hang on the velocity is constant so not only is it the initial velocity it's in the x direction because it never changes it's constant it's constant slowly so we've got some x velocity that never changes x dot I won't even put an i on it because there's no i there's no f there's no x dot one two or three I have some initial y velocity at the same instant I have some x velocity two vectors together because they both happen at the same time and they're both happening to the same object notice I didn't add their magnitudes together I added the vectors together what is this creature this vector sorry resultant yeah but that's just the word that's any two vectors you add together to get a resultant what is this resultant nope I added a velocity vector to a velocity vector I wouldn't add a velocity vector to an acceleration vector because we don't add things together that don't go together I wouldn't add a position vector to a velocity vector that doesn't make any sense I can only add two vectors together that are the same type of vector a velocity vector plus a velocity vector gives us a velocity vector what velocity what is this nope let's draw a bigger picture see if we can figure out what we've got here's that's the screen image we had yesterday the guy throwing the ball across the screen will ignore the part whether we're in his hand so we've got that pretty much as our screen image the video image how do you get a ball to do something like that if you throw it straight up it'll only go straight up and come straight down if you throw it straight out it'll only go out and then drop how do we get it to do this if I gave you a ball right now and said throw it so it does that you do what the guy on the screen did you take it you swing your hand back a little bit and you swing your hand forward you would give it a velocity something like that wouldn't you? a little bit up and a little bit out isn't that exactly what you'd do if I asked you to throw a ball across the room I hope that's what you'd do too athletic for some of you that's immaterial if you wanted to log like that you'd still give it a little bit up of course you'd do it over an hand because only girls throw under an hand I know what you're thinking that's what's going through your mind Samantha if you got something you can bounce it off the back of his head under hand or over hand I don't care isn't this what you do to get it to log across the room without even thinking about it that's what you do and you guys are good at doing stuff about thinking about it this is well let's let's see maybe we could even put a little O on it just to remind ourselves that that's not constant because the vertical component of that velocity is always changing but the horizontal component never changes that's interesting so we've got this vertical component that never changes no matter where that ball is it's got that same vertical component why is that there's no reason it should change the only way velocity is going to change is if there's some acceleration and there isn't so the horizontal part never changes and that's what we see pretty much here the line is straight because the velocity in that direction is almost perfectly constant almost perfectly that's a pretty darn good straight line the vertical component does change notice that the slope of this line is getting a little bit less steep as it goes to the top well that's what balls do they rise they don't keep rising as fast they slow down until what's the slope here? zero, what's the vertical velocity there? zero, it comes to a stop just for a split second and then notice the slope starts increasing negatively it's falling faster and faster and faster as it drops starts out with a pretty healthy initial velocity loses that until up here it has no vertical velocity the earth sucked everything out of it except it still has that horizontal component and then it starts to pick up a little bit more vertical velocity as it falls a little bit hard to draw with sidewalk chalk but that's the deal what's going on if I remember in the book this very very same thing showing constant horizontal velocity constant vertical acceleration actually it's constant acceleration for both it's just over here the horizontal side of the constant happens to be zero one of my favorite constants that little piece right there that's well we read it right off of here that's the initial exposition that's really tiny zero two it's just a consequence then remember averaging all these values it's a little bit of a piece that just happens to be left over alright I'm going to clean up here and we're going to summarize some of what we got with the rest of the details so get to a clean part on your page leave the semi permeable membrane there we are again I'll leave that little picture up the two main things we know remember the acceleration in the horizontal direction is zero because that horizontal position graph was a straight line it's not something I'm making up if you saw it yesterday if the if the acceleration is zero the velocity is I'm not making that up either that's that's the deal with with derivatives in fact I can even say a little bit more about what that constant is I've got a vector here representing the launch velocity whatever it's magnitude happens to be and it's at some launch angle if we had a cannon that's what you do you take that cannon and you point it up at some angle fire the cannon give you some launch velocity we bomb Hudson Valley Community College out of existence so this constant velocity we can even cast in terms of the launch velocity what would it be has to do with the magnitude of the launch and the angle of the launch what's trigonometry what would it give us for this component here on this side if that's the angle we know it'd be v0 what? not equals v0 cosine theta so if I gave you a gun that has a known launch velocity which any manufacturer or a riflery or whatever will tell you what the launch velocity is you take that gun or a cannon or what it is hold it up at a certain angle that's exactly what you do and you know then what the launch velocity was in terms of a vector it's magnitude and it's direction right there oh hey we know we know what that value is that was the 2.91 your numbers might be slightly different I think Samantha's will certainly be different it depends on whether you pick the same start point as I did when it was free from his hand let's see what else oh it's constant acceleration yeah constant acceleration happens to even be 0 the constant acceleration equations all apply they apply for any constant acceleration problem but if you look at those every place where A appears in those constant everywhere the acceleration appears in those constant acceleration equations you set A to 0 three of the equations completely disappear there's nothing left only one of them remains and it's delta X equals X dot T where T is the amount of time that's in the air that's all that's the only constant acceleration equation left all the rest are gone where delta X is how far it travels in the horizontal direction the whole show on that side X X there are no Y's over here they're just not welcome we're not even going to send them an invitation they're not welcome here there's a semi permeable membrane and Y cannot get through that membrane there's a Y here it's going to bounce right off no Y's vertical side let's summarize that what's the number one thing that's true about the motion in the vertical direction what drives everything in the vertical direction acceleration is constant and due to gravity so Y double dot equal minus G equals constant itself never has a negative sign that negative sign is only there because I arbitrarily chose it to be G itself is always considered a positive constant because that's not arbitrary God decided that that day 6,000 years ago somewhere in the Iran-Iraq area it was the first word there it was Monday Sunday he rested then we got to work G itself is not arbitrary but the minus sign is since acceleration is constant then any of the constant acceleration equations apply let's see let's look oh initial Y velocity since the acceleration is non-zero then the velocity in the vertical direction is always changing but it does have an initial value whatever it had that incident left that guy's hand there it is right there there it is in the picture right there what's it equal to here it was in the horizontal direction the initial velocity in the horizontal direction what's the initial velocity in the vertical direction give me more, give me the whole thing velocity is 0 it's the initial launch velocity but only that part in the vertical direction which is sine theta it's not a constant it's subject to acceleration in fact it would be one of the acceleration equations oh wait no I don't want to do that sorry the initial velocity yes it's a constant because once it leaves its hand the initial velocity can't change it's done it's already there it's what puts the thing in the air but the velocity from then on changes in a very predictable way changes in that way so we can even write down what it is as time goes by the velocity initially decreases vertically because the acceleration is down and then it starts accelerating down because the acceleration is down that in fact is one of the the constant acceleration equations just one of those in a slightly different form to fit this problem here and any of the others all constant acceleration equations apply I don't want to write them all down you've got them there in the sheet if you don't have that sheet if you lost it it looked like you had a quizzical look on your face or if you missed it that day I passed it out no trouble go to angel copy keep that copy for next year too if we have dynamics if I offer dynamics next year because we're using that exact same sheet all the constant acceleration equations apply it's just you need to go look at them anywhere there was an S in fact let me put up your sheet Joe if you don't mind there they are anywhere there's an S there's an A put a Y double dot or a minus G if you want just take those and cast them in the vertical direction pick out the S's put in Y's you got the vertical direction no X's no X's allow because they belong over here they can't cross this membrane either you can solve on this side acceleration equations cast in the vertical direction anything we need to figure out here well the whole thing's there there's nothing else to write down I got the whole thing down there a little more work to do here just because there's more going on since we have a non-zero acceleration so here's the big question of the day I had you put down the middle of your paper a semi permeable membrane because some stuff can go through that membrane what can and what cannot well we already know what cannot no X's no X dots, no X double dots no Y dots can go that way no Y double dots can go that way look it we don't even have a Y in the word we don't even have an X in the word we planned that out when when God made the English language which was of course the first and only language ever really used in the universe any American will tell you that what can go through this membrane one two three holes and only three things can fit through the membrane from one side to the other I heard something time can notice there's a there's not that T that T time it's the same time because those two things are happening at the same instant whatever velocity it has it's the same one for each instance so time can go either way in fact there will be problems you have to do where say the question is how far does it go in the X direction if we're going to shoot Hudson Valley Community College we need to know how far away it is so we need to know what the range is well to know what the range is to match it up with our velocity we need to know the time it's got to be in the air we may need to find that out from this side of the equation solve for the time bring it back through the hole and then use it here for the for this part of the calculation work okay what can get through the second hole there's a theta there and there's a theta there theta can make it back and forth as we need we have a couple problems in this homework set if I remember where you have to figure out what the launch angle is which is exactly what we have to do I guess if we were going to shoot Hudson Valley we know how far away it is we know what our our muzzle velocity is on the gun we have to figure out where we point the dang thing so we have to figure out what theta is we need to use it on both sides to make things work we have to figure out the velocity make sure that's the right distance all those kind of things we can do also if I remember on those homework problems where you have to find the launch angle the actual solution of those is a little bit tricky we'll set up the equations on one side and on the other and you've got to solve for theta and it's not a real easy thing to do but I'll talk about it when we get a little farther along with it how do you solve for things where the algebra just isn't very simple that's all good there's solution techniques when the algebra is not simple you can do quadratic equation and you can do solve for the unknown and all that stuff all the problems are just a little bit more difficult so I'll give you some tricks and techniques that will work one whole left ah, I plugged it the launch velocity what if we're talking about artillery or riflery or something is termed the muzzle velocity the speed with which it leaves whatever is shooting it whether it was the speed with which it left that guy's hand or the speed with which a bullet is out of the barrel of a gun whatever it might be if we're shooting a trebuchet what do you do? sling it sling it everybody know what a trebuchet is? oh yeah and that's all anybody builds nowadays the number one project for people with idle time and money those three things we need on both sides but launch velocity occurs at the same instant, the initial instant and it determines just what the rest of the flight's going to be as well as the launch angle does and then of course it's all happening at the same time just exactly what we found out from our data so I'm not making this up everything else I've taught you I make up as I go along but not this believe that nah it's in the book other little things you might know that are all oh by the way this is good for any projectile motion whatsoever I don't care what this launch angle looks like doesn't matter if it's up it's straight up it's straight over or even if it's down doesn't matter because it just gets fixed you handle it in here anyway doesn't matter if the origin is on the edge of a cliff and we're trying to shoot something below that cliff doesn't matter because we'll have a change in position here from one of the constant acceleration equations just put the number in and it all works out so this is in particular to this projectile motion or to any other projectile motion this is all true for every single projectile motion problem we could come up with you just have to get the right pieces in there where they go don't forget that some of these especially over here have minus signs on them if we're going from a high place to a low place delta y is probably going to be negative if we set it as such so make sure that's in there if we shoot to a higher place then delta y would probably be positive if you keep that negative sign I suggest you do it so that every projectile motion problem always has the minus sign in there and we never have to fuss with it but again it's arbitrary some of you are rebels a question before I clear the board and talk about one last little piece sometimes there's a particular part of either particular parts or particular types of projectile motion that can help you solve a problem if you remember for example I guess we've already seen it if we had any type of projectile motion some launch velocity doesn't matter if it's pointed up pointed horizontal pointed down or pointed straight up there is a point at the tippy top where there's no vertical velocity left it's all been sucked out and nothing mixture has been added for the way back down a lot of the problems I'll ask you something about to what height does the projectile go even if it keeps going the problem I say to what height does it go you need to recognize that at the top y velocity is zero remember for any acceleration equation there's three things you know one thing you don't to tell you which equation to use that might be the thing you need to know to find the thing you don't so even if the projectile keeps going doesn't mean you can't look at just the part to the top for your you can break the problem into its pieces like that if you need to and I believe we do have one of Alan didn't we have one of those questions you were asking about yesterday you need to remember at that point we do know something more that may not be expressly said in the problem may just say at the top you have to remember at the top we have no vertical velocity that's what defines the top almost done here one last thing there's also something extra there's also extra stuff you can know when you have a symmetric projectile shot one where it starts and ends at the same height launch velocity might give us a nice picture like that a couple extra things you that might help you in these questions one is that this peak this point where there's only horizontal velocity it comes right at the halfway point that can help if they ask you to find the time in the air and it's easier to find the time to the top find it and then double it or vice versa maybe it's easier to find the entire time cut it in half and you know the time it took to get to the top because it's symmetric also true not only is the distance half the time is half both of those are true the horizontal distance is half the time is half to the peak one other thing that's not quite as obvious this initial velocity over here is the same initial final velocity over here the magnitudes are the same and the absolute value of this angle is also the same that's not as obvious but it does come from the symmetry of the problem projectile motion you can effectively attack your enemies so go out there and make any aterondack proud PCC they're open range we can get them we'll leave them alone though because they don't have an engineering science program so they're no friends what whatever we need now