[John:] It's gotten off of me.
[Ruben:] That good is it?
[John:] [LAUGHTER]... Now. that away. Have you come across anything think ooh we could do with a look at that?
[Ruben:] Erm... no. [LAUGHTER]
[John:] Oh.
[Ruben:] Erm
[John:] Okay. Are you still working through past papers?
[Ruben:] Not really. I mean she, actually she might have given us.... Just been given a new exercise book, textbook.
[John:] So you've been, you've been back at school since I saw you last, and you've just
[Ruben:] Yeah.
[John:] been working on What?
[Ruben:] Graphs.
[John:] Graphs.... important thing about graphs.
[Ruben:] Erm.... Are they?
[John:] If there was an exam question which said sketch a graph of, where would the marks be found?... What would you get marks for?
[Ruben:] The axes.
[John:] Okay, X and Y.
[Ruben:] The... points
[John:] Which parti particular points?
[Ruben:] Points given for... like co-ordinates.
[John:] Erm let's, if they say sketch a graph,
[Ruben:] Right.
[John:] I they wouldn't give you any points.
[Ruben:] Erm... directions. Or shape of the line.
[John:] Okay, general shape. What would be the interesting parts of the shape?
[Ruben:] Where it crosses through the X and Y axes.
[John:] Okay, X and Y axes, anything else that might be an interesting point?...
[Ruben:] Erm
[John:] .... Just checking it was on at a reasonable volume.
[Ruben:] Erm... end... or anything else that might be interesting.
[John:] Mm.
[Ruben:] Where it cross the X and Y erm...
[John:] What's... where are the interesting points on that?
[Ruben:] Er er er... [whispering] there []....
[John:] Okay and these are, these are fairly interesting as well.
[Ruben:] Just.
[John:] What's happening in the, near the, the limits?... As X becomes a very very very small number, or a large negative number, if you like to think of it that way, where X is very large. So, sketching a graph is not the same as plotting one.
[Ruben:] Mm.
[John:] Any idea what that graph would be?...
[Ruben:] X squared.
[John:] Erm, what would... not quite the right shape... what would that graph be?
[Ruben:] Y equals something... plus X squared.
[John:] Right, there'd be some X squared in it. And there might be, it would be something X squared, and there might be something X and er well there's definitely a constant
[Ruben:] Mm. Yeah.
[John:] cos when X is zero still there's Y has still got a value. So that's going to be minus something
[Ruben:] Mm.
[John:] so it's about minus three.... X squared... cuts the X axis in two points.
[Ruben:] Two points.
[John:] This one
[Ruben:] Three points. It's
[John:] So?
[Ruben:] X cubed.
[John:] So this is, there's going to be an X cubed term in here somewhere, and there might be X squared and Xs and anything else, but if it cuts it three places, it's got three r three roots. So it's X cubed. What about this one?... I have to sketch curves This sort of thing. Just touches....
[Ruben:] Y equals X squared.
[John:] Something like, it's going to have an X squared in it somewhere. It's a bit lopsided, but it's
[Ruben:] Where's the X, where's the Y? It's gonna be about plus three, three plus.
[John:] Mm.
[Ruben:] X squared.
[John:] So it'll be something like... X minus three X squared.
[Ruben:] Mm.
[John:] if that's three. Just touches. Counts as two coincident roots. Two which are the same. So X equal three or X equal three.
[Ruben:] Yeah.
[John:] sort that one out.... Erm don't forget these endpoints. If you have something like... Y equals X squared plus six... What would that look like, going as X goes from minus infinity through zero to plus infinity? What would be the interesting points on that?
[Ruben:] Erm that'd be six. Y would be six,....
[John:] Mhm. nought is six.
[Ruben:] Erm
[John:] What about when X is about sort of point one. Nought point one. What's Y equal to, roughly? Approximately.
[Ruben:] Just over six, six point
[John:] s
[Ruben:] two.
[John:] Six and a bit. Erm that's when it's minus nought point one four, or plus nought point one I went through the minus point first. It's still about six, six, six and a bit.... What about when X is sort of around infinity? What does it approximate to?
[Ruben:] Erm minus infinity.
[John:] So, it's minus infinity or plus infinity. When X is very large or very very small.... at minus a million.
[Ruben:] Erm
[John:] What would that approximate to?
[Ruben:] A million plus six.... No...
[John:] Do you enter a big competition, and you could win a million pounds plus six. Compared to the million pounds nobody bothers much about the six, so
[Ruben:] Just a
[John:] when it gets very large, Y is roughly equal to X squared. So if you know the graph of Y equals X squared which unfortunately I can't draw tonight it's... draw it for you, could draw it from the inside keep telling everyone else. Right. Now, there's a Y equals X squared.
[Ruben:] Yeah.
[John:] So around here, round this bit,... there won't be any difference.
[Ruben:] Hmm.
[John:] Around... six plus six, let's say plus six in there.... For, for some length, that'll be sort of almost flat, won't it? I mean
[Ruben:] Mm.
[John:] till we get to X equals a one and it starts making a bit of difference, so it'll probably go like this. Okay, so a lot of it, that's a very important, those are sort of very important parts of the graph. Tend to
[Ruben:] Mm.
[John:] get neglected. People say oh I'll take it from minus five to plus five.... Right? If we had something like Y equals erm X minus ten. Right. Times... X minus erm or times ten minus X... like that, okay. Funny way of writing it, but you ca you could write it that way.
[Ruben:] Ten X.
[John:] Erm... people might say oh well we'll take it from minus five to plus five.
[Ruben:] Yeah.
[John:] And you'd get some sort of shape. But you'd miss a lot of important points.
[Ruben:] Yeah.
[John:] Erm if they took it from minus ten to plus ten
[Ruben:] Yeah.
[John:] you'd see a lot more things of interest and maybe say oh I'll take it from minus fifteen to plus fifteen. But you're still missing out a big, very important part of the graph, because for a lot of its range it actually, you know, looks something like well what would that look like? When X is gigantic.
[Ruben:] ... Ten times.
[John:] If, if X was a million say, you could forget about a million minus ten, let's just call it a million,
[Ruben:] [sniff]
[John:] times ten
[Ruben:] Ten.
[John:] minus a million, we'll forget about the ten, just think of the minus a million. So it's a million times minus a million. Roughly. A million times minus a million?
[Ruben:] Ah. Minus.
[John:] What's, okay, let's come down a bit. What's plus three times minus three?
[Ruben:] Three.
[John:] What's plus three times plus three?
[Ruben:] [whispering] [] Nine.
[John:] Right.... And, minus three times minus three?...
[Ruben:] Minus nine.
[John:] No. Minus three times minus three?
[Ruben:] Oh nine.
[John:] Right. So,.
[Ruben:] It's a million so one million
[John:] It's a
[Ruben:] times minus one million.
[John:] Is?
[Ruben:] [whispering] million times []
[John:] And what is the sign of it though?
[Ruben:] Minus. It's the same
[John:] Right. So it's minus a million squared. So wh when X is very big well, say, when it's big compared to ten
[Ruben:] Yeah.
[John:] this looks like Y equals minus X squared....
[Ruben:] Mm.
[John:] And this part of the graph, in there I mean that wouldn't start showing until we get beyond about say twentyish.... At that point, it could be doing all sorts of weird things, and you'd think oh, I've got this wonderful graph and I plotted a lot of points close together, so I know my sketch was right, because it looked like this funny W thing or M or something
[Ruben:] Mm.
[John:] erm it might well look like that for that very small piece of graph, but the overall picture you've lost completely by not going far enough so you, you want minus infinity to plus infinity. When you substitute infinity in, just think well if it was about a million, what could we ignore? I mean, you could work all this out... right.... Ten X plus another ten X. Okay? And then minus a hundred and minus X squared, and put the millions and things in but don't need to. You've got it factorized like this, you could just, oh, that's about a million. It's about minus a million, so it's about minus a million squared. And there'll be marks going for. It's where most people throw it away on the sketches, they just pick a tiny, tiny portion and look at that in great detail, say ah, look at that, wonderful.
[Ruben:] Mm.
[John:] But you've lost the, the overall shape of it. Tt, erm plotting graphs, you had one question didn't you where they asked a gradient?
[Ruben:] Yeah.
[John:] And you measured it with a protractor,
[Ruben:] Mm.
[John:] it was forty five degrees so you thought it was one, but the scale was two to one.
[Ruben:] Yeah.
[John:] The gradient is the tan of the angle, but you can only read that off the graph if you've got the same units. Same scale on the X and Y. Erm what would be... let's say someone gave you this... find an approximation for the gradient... erm Y equals X squared... minus one. [LAUGHTER]
[Ruben:] X squared?
[John:] Y equals X squared minus one, at around the point where X equals three. Someone wanted to know the gradient.
[Ruben:] So, so
[John:] Ooh.
[Ruben:] you d you'd have to find Y.
[John:] Okay.
[Ruben:] So Y is three squared minus one.
[John:] Yeah okay.
[Ruben:] be Y would equal eight.
[John:] Yeah.
[Ruben:] Erm... so
[John:] Good. Right, draw a sketch, cos we need to know what we're working with here....
[Ruben:] So Y could perhaps equal three....
[John:] the rest of the graph...
[Ruben:] ... the opposite point.
[John:] Right. Good. It's symmetrical down the Y axis....
[Ruben:] And the lowest point. [whispering] [] It'd be about one.
[John:] About one.... What's the graph going to look like here, when X is very big?
[Ruben:] . X is going to go up when Y is a million.
[John:] X X is a million.
[Ruben:] X is a million.
[John:] What, what would Y be?
[Ruben:] A million squared.
[John:] Yeah. So th the minus one doesn't make a lot of difference, so as X is very, when X gets very big, it'll just look like Y equals X squared, taking one off if
[Ruben:] Mm.
[John:] you're working with millions of millions. So it looks like Y equals X squared, moved up a bit, more or less, as a good approximation.... I mean a hundred is probably big enough, twenty five might be is more or less big enough.
[Ruben:] Mm.
[John:] Er five, say we can have five here, you can have twenty five, over here somewhere,
[Ruben:] Mhm.
[John:] behind your shoulder. Yeah it'll actually be twenty five minus one,
[Ruben:] Minus one.
[John:] but it's a good enough approximation just to see what it'll look like.
[Ruben:] So how do you work out the gradient?
[John:] Well you see, it's pretty flattish isn't it? The gradient of a curve is the gradient of tangent, so we can draw a tangent where you wanted it at that place. It would be the gradient of the tangent.
[Ruben:] Mhm.
[John:] If it was at the bottom where it's flat, where you just, you're just coming down, and then you started to go up again, just at the bottom there, it's, it's zero for a moment. But what would the gradient be there, and how would we find it?...
[Ruben:] Er...
[John:] It would, it, up here.... So could you do an approximation, a ver just a very rough guess. I mean does it, does it lie over that way or has it sort of gone back on itself? Lies over that way?
[Ruben:] Yeah.
[John:] Is it more or less straight up? More or less horizontal? About forty five or?
[Ruben:] It's probably about seventy.
[John:] Right. Maybe seventy, maybe eighty. So you could draw a rough sketch like that,
[Ruben:] Mm.
[John:] just measure it with a protractor, or measure along here.... That would be... if we drew wh wh how do we find the gradient really? We draw a little triangle, don't we?
[Ruben:] Yeah.
[John:] So at point one, it's nought. What would, what would that be at the point two? For X
[Ruben:] Three.
[John:] equals two? Two three. So we could get the point two three, and we could take the gradient of that as being fairly close to the gradient of the curve.
[Ruben:] Mm.
[John:] Erm how can we get it more accurately reflecting what's happening there? Right. Point something. So if we did something like that that's where we want it. That's where we want the gradient
[Ruben:] Mhm.
[John:] and we took a line from there to there, where this is that's where X is equal to, that'll go far. Nought nought point nine say.... And just the other side of it, where X equals one point one.
[Ruben:] Mm.
[John:] We're going to have, difference along the X axis will be nought point two.
[Ruben:] Yeah.
[John:] Find out what the difference, what the, don't forget the difference
[Ruben:] [whispering] On the Y axis [].
[John:] on the Y axis.
[Ruben:] So. That's point two there....
[John:] So what's, what's the value of Y going to be?... When X is nought point nine?...
[Ruben:] Point nine squared, minus one.
[John:] Okay. We can forget about the minus one because we're going
[Ruben:] Mm.
[John:] to take the minus one off again on this one so we've got, the difference is going to be X one squared minus one minus X two minus one is X two squared minus one, minus X one squared minus one. And they're just going to
[Ruben:] Cancel.
[John:] cancel out.
[Ruben:] one.
[John:] So the difference is going to be one point one squared minus nought point nine squared
[Ruben:] Point two squared.
[speaker003:] Is it?
[Ruben:] [whispering] Two point two [].
[John:] Can you remember the factors of A plus B times A minus B....
[Ruben:] A squared... minus A B
[John:] Yeah. A squared minus A B
[Ruben:] [whispering] Plus []... Plus B A.
[John:] Minus A B plus A B?
[Ruben:] two A B.
[John:] No. It's not two A B, it's minus A B and plus A B
[Ruben:] Plus A B.
[John:] which cancel each other out.
[Ruben:] Yeah. Erm
[John:] Then... plus B times A m
[Ruben:] Minus B squared.
[John:] minus B squared... One point one squared
[Ruben:] Minus
[John:] Point nine squared.... A squared minus B squared so it'll be, be one point one, plus point nine
[Ruben:] Minus.
[John:] times one point one.
[Ruben:] Minus.
[John:] Minus?
[Ruben:] nought point nine.
[John:] Nought point nine. I mean you could square them on your calculator, it might be easier.
[Ruben:] [LAUGHTER]
[John:] give you twice
[Ruben:] Two times two.
[John:] nought point two. Nought point two squared would give you what?
[Ruben:] Point four.
[John:] Point four. Try it on your calculator....
[Ruben:] Point nought four.
[John:] Point nought four which is s... very little.
[Ruben:] Mm.
[John:] But twice point two,
[Ruben:] Give you point four.
[John:] will give you point four. A hundred times as much.
[Ruben:] Mm.
[John:] So here we've got
[Ruben:] This is less than one. squared.
[John:] squared it gets smaller and smaller. So that's going to be nought point four.
[Ruben:] Mm.
[John:] So the gradient at that point is about two. Erm what wh where were we where did I say find the gradient? Where X equal to three?
[Ruben:] Yeah.
[John:] Sorry something wrong that.
[Ruben:] [LAUGHTER]
[John:] [cough] So we'd be, oh, okay, that's good cos you can do it.
[Ruben:] [LAUGHTER] When X is three?
[John:] So when X is two point nine, say. Two point nine, and then
[Ruben:] Y
[John:] when X one see what Y is down here.... It was three point one. Don't forget you can look at this, Y equals X plus one times X minus one, which, when you see it that way tells you where it's going to, where its roots are. Where it's going to cut the axes.
[Ruben:] Yeah. X... So there Y will be two point nine plus times two point nine minus one.
[John:] Right.
[Ruben:] Which is three point nine times nought point nine,
[John:] Okay.... But f we can, we can, we can save that to the end because... I think that was
[Ruben:] Aye.
[John:] confusing probably wasn't it? When we're doing it this way, we can forget about the, with this X, X two is going to be three point one squared
[Ruben:] [whispering] Minus one [].
[John:] minus one. This one is going to be two point nine squared, minus one.
[Ruben:] Yeah.
[John:] Subtract them that way round.... They'll go out. And we've got three point one squared, minus...
[Ruben:] [whispering] Two point nine squared [].
[John:] [whispering] Two point nine squared [].
[Ruben:] Mm.
[John:] Which will come to, if we add them it's six times nought point two. The gradient will be one point two. That's a bit better because a gradient of one is forty five degrees and we s we said it should be more than that. Erm... one point two. Find out which ang what angle has a tan of one point two.
[Ruben:] [whispering] Tan minus one [].... One point two.... Fifty.
[John:] Fifty. So you were, you were closer to seventy. You see how hard it is to see just from the graph, from a, from
[Ruben:] Mm.
[John:] a sketch, if you plotted it on paper, but this is without plotting it, we're getting it quite accurately. Erm... maybe someone says it could be useful for you to try it again, without any help from this page at all.
[Ruben:] [LAUGHTER]
[John:] So we've got Y equals X squared, minus one.... And we want the gradient at the point where X equals three. But we want to get very very close to it, like only nought point nought one before it and nought point nought one after it. So what value would? Around the point where X is three.
[Ruben:] Erm so you'd need... nought in the middle.
[John:] No, I'm not thinking of drawing a sketch at the moment, we've done the sketch on the other page
[Ruben:] Ah.
[John:] and we're just thinking of... what value are we going to, what would be the value of X point nought one before three?...
[Ruben:] Er two point nine nine.
[John:] Okay. Two point nine nine would be one of them.... Just after three would be?
[Ruben:] Three point nought one.
[John:] Right. Right?
[Ruben:] And in between would be three.
[John:] Between would be three. Up one that, we'll just take. What will Y be when X is that?
[Ruben:] Erm two point nine nine squared minus one.
[John:] Okay. So you'd just write it as, I mean as it's minus one, you can take it off as you go along, but it's, sometimes it'll be a minus six or something, so write it as two point nine nine squared minus one.
[Ruben:] Erm three squared minus one.
[John:] Okay. Now subtract
[Ruben:] when you have to
[John:] that Y from that one.
[Ruben:] Okay.... It's two point nine nine plus three point
[John:] Mm. Oh Oh, yeah, don't, don't do it all in your head.
[Ruben:] Okay.
[John:] Do the first stage first. Take that expression away from that, and once you're left, what happens to the minus ones?
[Ruben:] They cancel.
[John:] Right. So what are you left with? In terms of the squares.
[Ruben:] Two point nine nine and three point
[John:] Which one's first?
[Ruben:] Two point nine nine..
[John:] Now we'll just take the
[Ruben:] That, three point nought one, minus
[John:] Right, so it's three point nought one squared,
[Ruben:] Minus the two point nine... minus Erm
[John:] What was the difference in the Xs?
[Ruben:] Point nought one.
[John:] Point nought two.
[Ruben:] [whispering] Point nought two sorry [].
[John:] So we put that underneath, on the bottom half of the fraction.... Okay. That's, that's... Now when you've got an A squared minus B squared, you can factorize that I mean you could just do that on your calculator, and do it in one go, but it's quite easy to do it as... three point O one. A plus B, times A minus B. Okay?
[Ruben:] So three point nought one plus two point nine is point nought two. No sorry, is five point six.
[John:] Is six.
[Ruben:] And the
[John:] the top is six.
[Ruben:] Six.
[John:] Times.
[Ruben:] Times three point nought one minus [whispering] two point nine nine [], which is point two. Equals
[John:] Six point oh two.... Now work out what it's equal to.
[Ruben:] Okay.
[John:] All over?
[Ruben:] One point nought two.
[John:] Right.
[Ruben:] Do they just cancel out?
[John:] Right. They do cancel out.
[Ruben:] Six.
[John:] Six. So the gradient
[Ruben:] [whispering] Six [].
[John:] Wh when you do it to sort of fairly accurately like that, it six. When we did it fairly roughly, it came to one point two?
[Ruben:] Yeah.
[John:] Erm... six Because, because we didn't do this bit did we? That should have gone I think that was wrong. That should have been six times nought point two over
[Ruben:] Minu Yeah.
[John:] Which again comes to six. So, that's more like it. What's the, what angle has a tan of six, then? Now it'll be in the seventy or eighties.
[Ruben:] Eighty.
[John:] Eighty? Good.... to be fifty. Erm you can do that with any curve. If I gave you sort of X to the seventeen or something, well you could put two values in. And you could bring them fairly close together and say well I can't tell you exactly what the gradient is at that point, but if you draw a tiny little line and take the gradient of that.
[Ruben:] Mm.
[John:] That's er that's good enough, that's close enough. That's a very good approximation.
[Ruben:] Yeah.
[John:] So gradients are a very important part of, of sketching a curve. Erm you already pointed out some of the important points where the gradient gets to zero. Erm... does, does things that might be interesting?... Okay. You've already mentioned it earlier, actually, symmetry.... Not always symmetrical about this axis. They might be symmetrical about here or here. Or it this way round.
[Ruben:] What's that?
[John:] What's that? That's a. Erm
[Ruben:] Well I mean what gives you that?
[John:] What gives you that? Well, what does that look like from the way you're looking at it?
[Ruben:] s
[John:] The way you're looking at it, it looks something like erm Y equals X to
[Ruben:] [whispering] X []
[John:] the fourth plus three or something.
[Ruben:] Mm.
[John:] Good. So looking at it this way up I mean... it's... X equals Y to the fourth... plus three and that's all minus.
[Ruben:] [LAUGHTER] Mm.
[John:] X X equals three minus Y to the fourth.... It's a funny way of putting it.
[Ruben:] Yeah.
[John:] But you could write your equation that way. Er if someone says that's not good enough, I want it in terms of Y equals well okay we'll have to bring the three over to this side, and take the fourth root,. But you can always swap the axes, if you don't like what they give you.
[Ruben:] Mm.
[John:] So if, if I say draw erm... Y equals the square root of X, and you think I haven't got a clue what that looks like. You can square both sides. You can draw Y squared equals X, and all you have to do is say well, I can draw... the other way round.
[Ruben:] Mm.
[John:] Y equals X squared. You draw your Y equals X squared, turn the paper round that way, and whichever way, I'm going to draw Y equals X squared. And erm... Y equals X squared, and that's Y and that's an X. Then turn it the other way up to get X equals Y squared, and now that's the Y and this is the X. And this is a, should have drawn it that way round,
[Ruben:] Mm.
[John:] to get the, the X squared, this would be the minus X squared.
[Ruben:] Mm.
[John:] X squared look like that. Y squared equals X.
[Ruben:] Yeah.
[John:] Erm you don't have to do that very often, but it is sometimes, it fits. It's handy. Rather than doing it the other way. What, what have you been doing with graphs at school?
[Ruben:] Erm
[John:] Sketching and plotting?
[Ruben:] there's the revision.
[John:] . Have you done every one? Every question... twice... three times?
[Ruben:] Four, even.
[John:] Oh, that's too many.
[Ruben:] No actually,
[John:] Oh you'll have to [LAUGHTER]
[Ruben:] Oh I'm sorry, I'll go back to three.
[John:] you'll have to do them all you'll have to undo
[Ruben:] Do them or something.
[John:] That's it. Do them all backwards, yes.
[Ruben:] I haven't got it with me.
[John:] Mhm.
[Ruben:] Yeah.
[John:] Okay. So... you've been sketching them?
[Ruben:] Yeah.
[John:] And plotting them?
[Ruben:] Mm.
[John:] Have you been erm
[Ruben:] Speed-type graphs.
[John:] Mhm. Good. Aha, have you been asked, erm what is this? Here's a graph, now guess the equation?...
[Ruben:] No. [LAUGHTER]
[John:] No. Okay. You don't really need to, but it's, there usually isn't time to cover it, cos you've got other things, but it gives you a much better feel for it if you, if you see the other side of the picture all the time. That you're not just in isolation, looking at one little thing, because you can miss a lot.... Erm you might think that all graphs always look like that, or bits of graphs are always like that, or where does it all tie in?... Distance time graphs. One of those on every exam so far.
[Ruben:] Mhm. Yeah.... Yeah.
[John:] Are you happy with those?
[Ruben:] Er I think so yeah..
[John:] Okay. What's, what's happening here then?... This is a good way
[Ruben:] We do them in physics.... [door opening]
[John:] Oh lovely. Thanks very much.
[speaker003:] Okay.
[John:] [door closing]...
[Ruben:] I
[John:] Right, that's erm as time goes on along there,
[Ruben:] Mhm.
[John:] this is displacement, which is the distance from a fixed position. come from that way. bit of negative time in there. What's happening there then?
[Ruben:] Well it's speeding... up to there accelerating and it's
[John:] Mhm.
[Ruben:] It's speeding up again....
[John:] What's happened? That's, that's, that's a, that's a straight line, and you say it's speeding up? That seems to imply that, I don't think it's speeding up, [LAUGHTER] doesn't it []?
[Ruben:] Yeah.... Erm
[John:] So the gradient there, is S over T, how far you've travelled. Let's say these, let's say we mark this off in one second intervals along here. If I got the intervals right they'd be the same.
[Ruben:] [LAUGHTER]
[John:] So every one second, it's gone another ten metres or so.
[Ruben:] Yeah....
[John:] Gradient is the same all the time.
[Ruben:] Ah, so it's just steady
[John:] Fixed constant speed, constant speed. How about... have a bit so that bit...
[Ruben:] Erm... don't know.
[John:] Mm. S I'd say that's ten metres or something. Where are you after one second?
[Ruben:] Ten metres.
[John:] Ten metres. Where are you after two m?
[Ruben:] Ten metres.
[John:] Mm. It's not moving.
[Ruben:] Mm.
[John:] As time goes on, its distance stays the same. Now it looks like, ah just stopped moving again, straight line, fixed speed that, this looks as if it's accelerating. It's very, it's deceptive if you're not working out what you're doing and relating it back to the graph, and relating it back to physics, to an experiment you're doing, someone's checking the clock every second to see how much further it's gone. Oh well, in the last second it went another ten metres. Last second it went another ten metres. It's going at a steady speed. Fixed constant velocity. But here, that's still where it was last time I checked my watch.
[Ruben:] Yeah.
[John:] Not moving.... So that's a steady speed.
[Ruben:] And it stops.
[John:] That's, it changes... abruptly. From a steady speed.
[Ruben:] It doesn't need to brake or does it just?...
[speaker003:] Mm.
[John:] If it's a physical body, to change speed, it'll have to have to brake. And the s what would, so, that's a fixed speed.... There's another str another straight line.... There's another straight line, another fixed speed. Which one is faster?
[Ruben:] Every one second, ten metres. Every second that's going more than ten metres
[John:] Right.
[Ruben:] So that
[John:] So the steeper it is?
[Ruben:] The faster.
[John:] The faster it's going.... So at this point, it's almost flat. So it's going very slowly. Now what's happening to its speed?
[Ruben:] Accelerating.
[John:] Yeah. The speed is increasing. Sort of
[Ruben:] Stop.
[John:] sort of actually stops, and then it's going very slowly again for a bit, and then what happens here?...
[Ruben:] very, it just goes very fast
[John:] Right.
[Ruben:] in a very short space of time then, but then carries on.
[John:] Very short space of time it's gone quite a distance. So that was a, that is at, that constant speed there is much faster than, than this one. And then, it's stationary
[Ruben:] It's
[John:] for a while. And then
[Ruben:] Just goes down there.
[John:] What's happening here?
[Ruben:] Deceleration.
[John:] Well let's make, let's make this bit a straight line, from there to about there. That's a straight line.... So, well let's go back to the physics.... We've just been going on timing it for one minute or something.... And we time it again, one minute and one second...
[Ruben:] It's reversed.
[John:] It's reversed. How far away was it when I started the watch? Thirty metres away. How, how far is it after another one second? Twenty metres away. Ah. Ah! [mimics fast object] It's coming back. Okay. So that's going backwards. Most graphs will keep going on up, and there's a tendency for people to make them come back down again
[Ruben:] Mm.
[John:] or at least to where they started. That means it's going backwards. Erm... now you don't do, paper three, you don't do... velocity time graphs, but you probably do them in physics don't you?... You would do them in physics.
[Ruben:] It's just like pure coincidence.
[John:] Aha!... Oh that's a good one that.
[Ruben:] Revising it yesterday to be honest.
[John:] That's a good one. got all pictures in it. Yes it has, but these are, those are, I've seen those before and they are very nice graphs, I would recommend them to anyone.... Right.... So, you were revising them yesterday.
[Ruben:] Should I
[John:] Well the big thing, ah. See, it's very easy to get confused between distance against time, speed against time, and even acceleration against time.
[Ruben:] Yeah, because I mean it had them in our exam,
[John:] Right.
[Ruben:] in our mock, we had like those four pictures, and you had to say which was
[John:] It comes up time and time again, and you have to know the difference between the different types. So... you also, when you get used to each type individually, need to then be able to switch between
[Ruben:] Mm.
[John:] between one and the other.
[Ruben:] Yeah.
[John:] So if we've got erm... acceleration against time and we get, no, no peeping. No peeping. We get a flat one, what does that mean?
[Ruben:] It's Every one second, you've gone further.
[John:] We haven't gone anywhere we've measured the acceleration.
[Ruben:] The acceleration there is say forwards.
[John:] [whispering] One two []
[Ruben:] And for every one second, you're going forty metres per second. So you're going at a steady speed.
[John:] Per second. It's, it's a, it's an acceleration, it's metres second to the minus two. Now are you happy with that, because most people aren't? Okay. We'll come back to that in a minute, but... let's say... a rocket, or a car, which would you prefer that do?
[Ruben:] Car.
[John:] Car. Constant acceleration car. Erm... no frictional resis resistance, you just put your foot down and it's, you know where a car accelerates from rest
[Ruben:] Mm.
[John:] and then after a while the acceleration reduces? The speed is still going up, the acceleration, and eventually it reaches a top speed and it stays there.
[Ruben:] Mm.
[John:] Okay, it's zero acceleration, it's not getting any faster.
[Ruben:] Mm.
[John:] Erm... how would we measure that? A constant acceleration. Say we had an accelerometer. An acceleration measurer.
[Ruben:] Mm.
[John:] Which force on a weight or something can feel acceleration, I think you can, my son said you can't but I think you can
[Ruben:] [LAUGHTER]
[John:] So, acceleration is a constant. What's your eventual top speed?... It just, just keeps going faster and faster and faster and faster
[Ruben:] Infinity.
[John:] Infinity, speed of light, when you hit something. It's pretty high.
[Ruben:] Mm.
[John:] Erm now how are we going to measure acceleration?... Let's say we've got erm a radar for, for measuring speed
[Ruben:] Mm.
[John:] velocity. Okay. And we'll set up a, so we're going to, you, you devise an experiment, Neil. Measure acceleration. There's a car, and we want to know its acceleration between two points. [tape change]
[speaker003:] doesn't matter.
[Ruben:] There's your car. Erm... you start the stopwatch. Er stopwatch you m you time it from... there to the ten metre.
[John:] Erm what's along your X axis? Your independent variable?
[Ruben:] Time.
[John:] Time. So, when would you next check its acceleration? Would you measure in terms of distance or in, in terms of other time
[Ruben:] Time. Ten seconds.
[John:] Ten seconds. We'll make it one second.
[Ruben:] Okay.
[John:] Make it one second. Alright, make it ten. Sorry,, make it ten seconds.
[Ruben:] Plus one second.... Then the you have to... have y how far the car has travelled,
[John:] Do you?...
[Ruben:] know how quickly.
[John:] What is acceleration? Give me a figure, that you might have seen in an advert, when you're drooling over BMWs and things, Maseratis.
[Ruben:] Miles per hour.
[John:] Telling you about its acceleration. What does it say?
[Ruben:] Goes from nought to sixty.
[John:] Good. In?
[Ruben:] Six seconds.
[John:] Six seconds. Nought to sixty in six seconds. So those are the units of acceleration.... From one speed to another speed,
[Ruben:] In however qu
[John:] in so, so much time. So let let's say, instead of quoting from nought, they said well a much more useful figure is perhaps, which they do quote in lots of these car magazines, is erm forty to sixty
[Ruben:] Mm.
[John:] because that's what you need for overtaking someone. You can go from forty to sixty in point three seconds or something, you can really shoot past anything on the road. So forty to sixty, miles per hour, metres per second, whatever you like, let's say metres per second. Let's say it goes from... forty to si forty metres per second to sixty metres per second in ten seconds. More of a slug, this car. [LAUGHTER]
[Ruben:] [LAUGHTER]
[John:] So what would that acceleration be? How w what would the units of acceleration be?
[Ruben:] Twenty metres per second per second.
[John:] Twenty metres per second, per second.
[Ruben:] Per second.
[John:] It's simpler to say per second per second. It's Or per second squared, or second to the minus two. What is this rubbish, it doesn't, doesn't, cannot possibly mean anything real.
[Ruben:] Mm.
[John:] Well, it is real. It's... erm fifty miles per hour... per hour.
[Ruben:] Yeah.
[John:] Right well we'd normally say, miles per hour per second. What's, what's your acceleration at the moment? Well it's ten miles per hour per second. Okay? Well every second, I'm going I'm going ten miles an hour faster. So miles per hour per second, or you could measure it per hour if you wanted. Every hour, I'm going six hundred miles an hour faster
[Ruben:] So
[John:] so it's miles per hour per hour.
[Ruben:] you'd need a stopwatch and a speedometer.
[John:] You'd need a something to, I say, we've got something to measure the speed, radar gun,
[Ruben:] Mm.
[John:] not interested in the distance, just the velocity, the speed. How fast is that car going now? Okay, he's doing forty miles an hour. Measure him again in one second's time
[Ruben:] He's doing sixty. he's doing sixty miles an hour, so his acceleration was twenty miles per hour per second. Per second.
[John:] Twenty miles per hour per second, miles per hour, what's the change this miles per hour into metres per second. Okay, so, say a hundred and twenty metres per second per second.
[Ruben:] [LAUGHTER]
[John:] And that's where your per second per second comes in. Are you a bit happier with that?
[Ruben:] Yeah.
[John:] But not totally convinced. [LAUGHTER]
[Ruben:] [LAUGHTER] No [].
[John:] Erm... that length per time per time, is the unit. Let's just have a little look. What are the units of displacement?
[Ruben:] Er
[John:] What's the difference between displacement and distance?
[Ruben:] Displacement is how fast it's going, the speed. Miles per hour.
[John:] Show me something that is erm arm's length away from, from your right shoulder. Okay, show me something else that's an arm's length from That's your right shoulder isn't it. Arm's length. I was expecting you to go like this. [cough] Right, okay, that, show me something else that's arm's length from your right shoulder. Right. Up there
[Ruben:] [LAUGHTER]
[John:] Right. That's distance.
[Ruben:] Mm.
[John:] Displacement is distance in a specified direction. So if I said show me something that's arm's length from your right shoulder north, or up, or west or some way, okay, or on a bearing of thirty seven degrees [LAUGHTER]
[Ruben:] Yeah.
[John:] that's displacement. We don't usually tend to bother with distance much, so when, when the police measure your speed, they're actually measuring
[Ruben:] My dad's
[John:] your...
[Ruben:] my dad's speed.
[John:] your dad's speed.
[Ruben:] Not mine.
[John:] You don't drive?
[Ruben:] No, no, no.
[John:] Yeah, of course.
[Ruben:] Can't be my speed. Not unless I'm on me bi on my bike, and I doubt if they'd measure me. you, you did eighty on your, on your bike did you?
[John:] Over, over fifty on a pushbike. Do you know Bo?
[Ruben:] Yeah.
[John:] You know Ben Lech? hill, red hill going down to? I was young and foolish at the time,
[Ruben:] [LAUGHTER] [LAUGHTER] Fifty miles an hour on a bike []?
[John:] Very, very high gear on it, we used to go out a couple of us
[Ruben:] [LAUGHTER]
[John:] And er... we were passing this M
[Ruben:] You've got this on tape remember?
[John:] Oh no! [LAUGHTER]
[Ruben:] [LAUGHTER]
[John:] We went past this bloke in a Morris Minor,
[Ruben:] [LAUGHTER]
[John:] a convertible, with a eyes streaming, it was really hard to see, and clocked his speed at fifty, and er the bloke behind me nearly got hit, [LAUGHTER] because when I went past this bloke [] I went [swerving noises]
[Ruben:] [LAUGHTER]
[John:] all over the road.
[Ruben:] No you didn't really though did you?
[John:] [cough] Ooh no.
[Ruben:] No, no you didn't.
[John:] No officer I didn't.
[Ruben:] [LAUGHTER]
[John:] Distance okay?
[Ruben:] Yeah.
[John:] And displacement.... When they measure your speed, they're measuring your velocity really, they're measuring it in a particular direction, along that road.
[Ruben:] Mm.
[John:] Erm so... we'll call it distance, right? But we're actually talking about displacement, just so that you feel more comfortable with it, cos there are other things coming up to make you feel less comfortable you see. [LAUGHTER]
[Ruben:] Okay.
[John:] So the police say you were doing forty miles an hour. You say how do you know I was doing forty miles an hour? I wasn't driving for an hour, you weren't following me for either an hour or forty miles. They say, ah well, we followed you for a tenth of a mile, right, and it took you so long to cover that. And we, our little machine here has calculated it that that's forty miles an hour.
[Ruben:] Yeah.
[John:] Okay, so what did they do? They had your hundred a hundredth of a mile say, let's say it's erm... let's say it was a tenth of a mile. Nought point one miles. And they divided that by erm... nought point O
[Ruben:] [LAUGHTER]
[John:] hours.
[Ruben:] Mhm.
[John:] Right? And they said you did point one mile in point O O one hours
[Ruben:] Hours.
[John:] so what speed is that?
[Ruben:] Er
[John:] Have a guess at it then check it on your calculator.
[Ruben:] Point... [whispering] one divided by point zero one []. Will the two first cancel out?
[John:] Do they?
[Ruben:] No.
[John:] If I gave you this... tenth, hundredth, thousandth okay?... What would you do?
[speaker003:] [LAUGHTER]
[Ruben:] Run.
[John:] that's hours.
[Ruben:] Are those? That one and that one cancel.
[John:] It would cancel out. So you've got one over one hundredth. So that's a bit tricky that, doing it that way, you've got one over one hundredth. The, the easy way, do you like fractions on the bottom of fractions?
[Ruben:] No. Oh, times by... ah.
[John:] What are you going to times it by?
[Ruben:] Times that by ten over one. No, times that by a thousand over one.
[John:] Times that by a thousand.... Okay, and you've done, you've done that on one bit of the fraction, so you must do it on the top as well. Times that by a thou aargh. You've got to times that by a thousand. Okay? Multiply the top and bottom of the fraction by the same
[Ruben:] Mm.
[John:] thing. So why did you multiply that by a thousand over one?
[Ruben:] To get that
[John:] a thousand times a tenth. So doing a hundred miles an hour.
[Ruben:] [astonished snort]
[John:] Well, for a hundred miles or an hour, but you did a tenth of a mile in a, a thousandth of an hour.
[Ruben:] Yeah.
[John:] So, you did so many miles per hour, we calculated, you would have done so many miles if you'd been going for a full hour at that speed, so your dis your, your speed, velocity really but we'll call it speed, right? Your speed was so many miles per hour. okay. And then they measure your acceleration. Well, when we first clocked you, you were doing a hundred... miles per hour.
[Ruben:] Yeah.
[John:] One hour later, [LAUGHTER]
[Ruben:] Yeah.
[John:] you were doing [LAUGHTER] five hundred mile an hour [].
[Ruben:] It's you on your bike isn't it?
[John:] [LAUGHTER] It is []. Shh!
[Ruben:] I'm sorry. You're not on your bike.
[John:] I don't mind I don't mind confessing b up to fifty, but five hundred mile an hour,
[Ruben:] [LAUGHTER]
[John:] even in a, in a foreign country, like Wales
[Ruben:] [LAUGHTER]
[John:] it's a bit much. Okay. So, you went, you gained four hundred miles an hour
[Ruben:] In one hour.
[John:] per hour. You're accelerating at a rate of four hundred miles per hour per hour.
[Ruben:] Per hour.... Mm.
[John:] Mm.
[Ruben:] So that means in another hour, you'll be going nine hundred?
[John:] I in another hour, I'd be doing four hundred more than that, so nine hundred. Or, I mean this is if it's an average, and sort of an even acceleration.
[Ruben:] So before you were doing minus three hundred mile an hour.
[John:] Yes.
[Ruben:] Bet you didn't expect to minus, did you?
[John:] I, I, I haven't recovered from that yet.
[Ruben:] Yeah I know.
[John:] Negative speed.
[Ruben:] [LAUGHTER]
[John:] Yeah.
[Ruben:] You're reversing at minus three hundred miles an
[John:] So acceleration is rate of change of velocity,
[Ruben:] Mm.
[John:] which is something like speed, only it's a bit it's got a direction.
[Ruben:] Velocity's given a direction. Yeah.
[John:] Right. And velocity is rate of change of displacement. Which is sort of distance, but with a direction. The direction that velocity has got comes from measuring the distance, the displacement, in the direction. Right? So if we say he went from, let's, here's, here's a map. And there's north.
[Ruben:] Mhm.
[John:] Right. And let's say he went from there to there. A distance of two miles in one hour, this time we're walking.
[Ruben:] Mm. [LAUGHTER]
[John:] Let's make it half an hour, okay? Then
[Ruben:] [whispering] Speeded up [].
[John:] his displacement
[Ruben:] Is two mil
[John:] the amount he walked was, his displacement was two miles south. The distance he walked was two miles. But the displacement, how far is he from where he started? He displaced himself by two miles south.
[Ruben:] Two miles south.
[John:] Okay. So his velocity,
[Ruben:] Yeah.
[John:] if he did that in half an hour, what would his velocity be?
[Ruben:] It's just speed given direction, so two miles
[John:] Two miles
[Ruben:] per
[John:] two miles south, in half an hour.
[Ruben:] Two miles per, one mile per hour,
[John:] Mhm.
[Ruben:] per
[John:] Mhm. Is it?... If he did two miles in half an hour, how far would he go in one hour?
[Ruben:] Four mile.
[John:] Right.
[Ruben:] So it's four miles per hour... per hour.
[John:] Per, no, well
[Ruben:] Per
[John:] we're just talking about this velocity stuff,
[Ruben:] Oh, so it's four miles per hour.
[John:] South.
[Ruben:] South. So you have to put the south? Four miles er hour sou
[John:] If it's a velocity. If it's a velocity. They won't bother with it on G C S E maths, but physics they want to know the difference between velocity and acceleration. So, if he was going, if he, he's walking and he breaks into, a breaks into a trot,
[Ruben:] A Trot? Gallop.
[John:] A canter to start with let's say.
[Ruben:] Oh no, that's horse racing. I I don't want to talk about that.
[John:] [LAUGHTER]
[Ruben:] My horse won.
[John:] A National oh! Ooh.
[Ruben:] My horse won.
[John:] [LAUGHTER] Ooh. Oh dear []. [LAUGHTER] And did you have a lot on it?
[Ruben:] Twenty er
[John:] Twenty million?
[Ruben:] Twenty pounds was going on it. It was just in the sweep.
[John:] Oh.
[Ruben:] Sixty to one outsider, and I thought oh well, I don't stand a chance on it. And it won.... Is, it was my sweep that I was doing so I'm keeping the money, nobody else can have it. Since my horse won.
[John:] Well there was, did you hear about the woman in Ireland who won about a hundred and fifty thousand? Did you hear that?
[Ruben:] No.
[John:] Oh, she drew, she was the only one who didn't draw a horse on the National,
[Ruben:] Mm.
[John:] I think it were the Irish Times or something, or the Irish Betting Times or something like that, and she was the only one who didn't draw a horse, and she thought oh well that's it, no chance. And they, they, sort of the panel met, and they decided to give it to [LAUGHTER] her, because hers [] was the nearest to a result.
[Ruben:] What?
[John:] The other people who'd drawn a horse,
[Ruben:] Yeah.
[John:] well their horse
[Ruben:] This is a sweep? Yeah.
[John:] . Yeah, this was it was some form of competition, I didn't hear what
[Ruben:] Yeah.
[John:] it was, I just caught the tail end of it on the news, on the
[Ruben:] Yeah.
[John:] radio. And erm, but they decided that as she hadn't drawn a horse in the National, she'd got the n she was the nearest to a winner that they could
[Ruben:] Aha.
[John:] find, because all the other who had drawn a horse they, they were void.
[Ruben:] Yeah.
[John:] [LAUGHTER] So she won [].
[Ruben:] [LAUGHTER]
[John:] And she'd already decided, no chance I didn't draw a horse.
[Ruben:] [LAUGHTER] Cos she didn't draw a horse she won, yeah.
[John:] cos she didn't she won.
[Ruben:] Mm. Did you put a bet on?
[John:] Did I? I didn't.
[Ruben:] You're a mathematician.
[John:] I I often, well, usually, if I'm in the country at the time I bet on the National just for a, for a to sort of waste some money.
[Ruben:] But you've got to check out the odds.
[John:] I l I do check out the odds,
[Ruben:] Yeah I thought you might do.
[John:] and erm I only put place bets on. Last year I had a place bet on the winner, I couldn't even tell you what it was now. But I didn't bet it to, I didn't back it to win, I only backed it to come in
[Ruben:] Each way.
[John:] the first four. Not each way, just place.
[Ruben:] Yeah.
[John:] Just er wh when you put an, when you put an each way bet, you put two bets on at the same time.
[Ruben:] Mm.
[John:] One bet says that horse will be first. The other bet says that horse will be in the first three. Or if there are a lot of runners, that horse will be in the first four.
[Ruben:] Mm.
[John:] Now when you're, my grandfather, who was a serious betting man, always used to tell me, he was pretty good at it actually, he'd say [mimicking] no good each way bet. Betting against yourself is an each way bet, you're just giving money to the bookie [].
[Ruben:] [LAUGHTER]
[John:] I thought well you give money to the bookie anyway, but
[Ruben:] Yeah. [LAUGHTER]
[John:] so erm I just put a place bet on. And put it on, they used to give me tote odds, sometimes I asked for them, usually they'll automatically give me tote odds. Sometimes I've asked for them. Now the odds on the tote, I mean if you looked at the, I didn't bet on the National, cos I didn't like the odds.
[Ruben:] Mhm.
[John:] There was a race just before the National, might have been the one before, or the one before. But something like erm a five or six horse race. And they had seven to one, there was eight to one, there was a ten to one,
[Ruben:] Mhm.
[John:] National, a forty horse race more or less isn't it, thirty
[Ruben:] Mm.
[John:] nine or something,
[Ruben:] Thirty nine,.
[John:] thirty nine horse race, anything can happen.
[Ruben:] Yeah.
[John:] Absolutely anything can happen in the National, as, as was
[Ruben:] yeah.
[John:] proved this year.
[Ruben:] Mm.
[John:] And they had the, the erm they had sort of five out of the five or six horses that I thought were in with a good chance at eight to one. They were eight to one in the National. I, I'd take eight to one o over any of those brilliant horses sort of getting round once.
[Ruben:] Yeah.
[John:] I thought, these bookies are going to rake it in. So I was I wasn't disappointed then [LAUGHTER]
[Ruben:] Yeah. [LAUGHTER]
[John:] that they didn't. But I thought it was very interesting, Peter O'Sullivan didn't have a bet
[Ruben:] Yeah.
[John:] and he said it was the only year that he hasn't. I thought he knows something. There's some fix
[Ruben:] [LAUGHTER]
[John:] Er no, quite often, I mean there was seventy five million pounds
[Ruben:] Yeah.
[John:] on that, and it was a lot of money. And my old grandfather used to say too sometimes
[Ruben:] [LAUGHTER]
[John:] I'd say it, what do you think of this horse, and he'd say oh, it's carrying too much money.
[Ruben:] Yeah. Too much of a favourite.
[John:] Too, too mu too many people betting on it. Something, something or someone will get to it. So it'll come in a close second. And
[Ruben:] Yeah.
[John:] the odds will be such that it's not worth doing for a, a place, but if you're betting, I mean forget how much I got last year, but I think it came in about thirty to one or something. So I didn't get thirty to one, I got
[Ruben:] Mm.
[John:] a quarter of thirty to one, which is still
[Ruben:] Mm.
[John:] much better than backing it to win and it comes second. Anyway, it did win, and the wife was saying why didn't you bet it, why didn't you back it to win! Why didn't you back it to win. She backed the second each way, despite me saying I'm not, I'm not going in there [LAUGHTER] and back it each way for
[Ruben:] Yeah.
[John:] you []. Oh alright, alright, I will then. But erm, so she got erm she, he made a bit of money I think, I think I went rash and put a fiver on.
[Ruben:] It's just the fu I mean I, I'd ne
[John:] But it's when you're watching the race.
[Ruben:] I'd never bet on any ra I'd never gamble, I mean, I would as a joke, but like not serious. I mean, the National I thought, yeah, it's a good laugh like. And that's the fun of it, any horse could win. Anything could happen. Much like it did. But I mean my sixty to one horse!, never even been heard of. I mean the jockey, the jockey
[John:] Oh, I wouldn't I wouldn't
[Ruben:] was crying.
[John:] He would be. But, mind you, there were a few very good horses that d weren't
[Ruben:] Mm.
[John:] in it. [LAUGHTER]
[Ruben:] Yeah. I suppose
[John:] I, I, I wouldn't have given that a chance, I must admit I would never have backed it.
[Ruben:] Sixty to one,.
[John:] Sixty to one.
[Ruben:] I thought before it, well, forget that one. I had another good horse, and I thought well I'll stick with that one, I think that one might stand a chance. And I saw it, and they g and I saw it wasn't going to happen. Like, when they all went off and I thought well I don't see, why don't they just stop it. And like I thought oh well, I'll watch it like, and they're commentating anyway.
[John:] didn't know whether to you know, whether to commentary
[Ruben:] Yeah, I know, they're like.
[John:] Well, they're doing well there, it's all a waste of time I mean
[Ruben:] Yeah. Yeah.
[John:] obviously.
[Ruben:] And like, kind of I saw my horse, it took the lead near the end as well,
[John:] [LAUGHTER]
[Ruben:] it was like go on, go on, go on! And it won. And I thought oh well. [LAUGHTER]
[John:] It's a wonder that bloke didn't get lynched..
[Ruben:] Yeah, I know. But I think I heard on the television like somebody goes I'll see you in court, I'll sue you for millions so like, it's terrible, but I mean, they, Desmond Lyneham was int was erm tt talking to so to someone from Ladbrokes,
[John:] Mm.
[Ruben:] and he goes well that's a million to one chance that that would, and Des Lyneham goes I wouldn't have minded having [LAUGHTER] a little flutter on that [].
[John:] [LAUGHTER]
[Ruben:] And he goes you couldn't have got the odds on it like... putting a pound on like, I bet you no horses finish, you know. [LAUGHTER]
[John:] ... That wasn't five pound, I remember now. It was five one pound bets that we put on between us, I put two on, my wife put three on, and all mine were place bets, and I got the winner and I got the fourth, and my wife got the second, and the other one of hers lost. so we were alright. But mine were only a quarter of the odds they were paying at
[Ruben:] Mm. But still.
[John:] erm and hers were a quarter of the odds, plus she lost her stake on the part that was betting on them to win.
[Ruben:] Mm..... Yeah. But I mean I was thinking like as well cos like we did taxes like a few weeks ago didn't we, and the tax that the government get off that race, I mean seventy five million on the whole race,
[John:] Right. betting tax
[Ruben:] and then and then
[John:] is ten percent now, isn't it?
[Ruben:] Yeah, well I paid fifty pence per horse, on our five horses.
[John:] What stake?
[Ruben:] But I er it was about two f yeah, I put a fiver, but I put it each way, so
[John:] Five pound each way?
[Ruben:] Yeah.
[John:] So it was a ten pound bet?
[Ruben:] Two fifty. Yeah. No, it was three horses, five pound bet,
[John:] Mhm.
[Ruben:] on each. And it was fifty P. I think, I can't remember
[John:] I don't understand what your stake was.
[Ruben:] It was, I put five pound on each way.
[John:] What was the total total money that you, that you put on? Oh, they take tax off now, don't they?
[Ruben:] Yeah, or
[John:] Yeah.
[Ruben:] Aye you can either pay for tax when you pay,
[John:] Yeah.
[Ruben:] or take it off your winnings, but [LAUGHTER] I'm not going to take it off my winnings so []
[John:] Yeah. So I think probably about ten percent, so they lost
[Ruben:] Yeah.
[John:] about seven... seven and a half million.
[Ruben:] Yeah.
[John:] [LAUGHTER]
[Ruben:] Yeah.
[John:] Okay. Now, so these racehorses, let's draw a graph of the racehorse.
[Ruben:] It's not called actually
[John:] It's called erm What was the one with the erm Scudamore?
[Ruben:] Scudamore, erm Captain Dibble.
[John:] Captain Dibble. I fancied that. I thought it
[Ruben:] Yeah, I thought that was, I had a bet on that.
[John:] Captain Dibble's good chance. So there they are, ready all ready for the off,
[Ruben:] Mhm.
[John:] and they're not fidgeting, they're absolutely motionless,
[Ruben:] [LAUGHTER]
[John:] and they're at the start line.
[Ruben:] Yeah.
[John:] So time is zero. And displacement, how far they've got from the start line?
[Ruben:] It's zero.
[John:] Is zero. You put S for displacement. Which is why we don't use D for distance. Right. They're off. And it's over to Ruben at the, at the stand commentary box.
[Ruben:] Okay.
[John:] Well Ruben, what's the displacement like, tell us all about the velocity and the acceleration?
[Ruben:] [LAUGHTER]
[John:] How's it going?
[Ruben:] I'm waiting for I T V to get there, okay.
[John:] [LAUGHTER] Oh it's a false start []? [LAUGHTER]
[Ruben:] Yeah, yeah. Alright.
[John:] They've got negative acceleration. They're all going backwards.
[Ruben:] So for each one's like ten metres,
[John:] Now along here, is your time.
[Ruben:] Oh. So, ten seconds.
[John:] all of these all these graphs that we're thinking of have got time along the X axis. what's happening every ten seconds.
[Ruben:] So for every ten seconds there the horse... s its displacement is.
[John:] So it's a distance they've got away from the start.
[Ruben:] Ten metres.
[speaker003:] Ten metres.
[John:] ten metres.
[Ruben:] Okay, so in the first ten seconds, probably gone about
[John:] Let's say they go at. Wh what happens from start to
[Ruben:] They start off slow and get then faster and faster.
[John:] Okay, so the gradient... which shows you how fast they're going If it's more or less horizontal, they're still.
[Ruben:] Mm.
[John:] If it's just a little bit of an angle, they're walking. If it's almost straight up
[Ruben:] So it's going to be
[John:] they're doing two thousand miles an hour.
[Ruben:] It's gonna be
[John:] So it's going to go up like that, gradually Good
[Ruben:] And then
[John:] Okay, take it up like that.
[Ruben:] So...
[John:] Well, they false start, and now they stop.
[Ruben:] Oh.
[John:] Okay, they stop.
[Ruben:] They're decelerating.
[John:] They've, they've stopped, they're just standing, stood stationary not going anywhere.
[Ruben:] Yeah.
[John:] Right. Come back. They all come back to the start.
[Ruben:] So, it'd be the exact opposite.
[John:] Er they walk back.
[Ruben:] So ah.
[John:] They w they turn round and they walk back.
[Ruben:] So they'll be... ah. So it'll be just like
[John:] Yeah, nice. A gentle, a gentle slope, so they just walk back. Like that.... They stop. When they get back to the start again. In fact they go a little bit past the start.... And then they come back to the start again. And they stop there for a bit, when they've gone past the start.
[Ruben:] Okay.
[John:] go back to the start again.... Okay? So they're off. That, that's, that's more of a, and they're doing very little to start with, and then they rapidly get up to their top speed, and they level out.
[Ruben:] And then the man goes with his flag.
[John:] Whoa! And they stop. They usually don't stop dead, they usually dec decelerate, so what's happening here is I'll just draw it up here. Start off, the speed's nothing to start with.
[Ruben:] Mm.
[John:] Or virtually nothing. It's getting steeper, and then, speed gets slower until it stops again. So zero speed, flat, level, okay? Getting faster, steady speed in that bit, and then from the speed there which is steep, as you follow the tangent of that curve, it's slower, slower, slower, and they slow down to a walk, and then they stop.
[Ruben:] [LAUGHTER] Mm.
[John:] [whooshing sound] Okay that was the first one. Now this time, in this re-run, the second attempt at getting this National started, not having any flags, red white or anything else, used a proper old starting pistol, and we fire it in front,
[Ruben:] Shoot the horses.
[John:] and we just I'm sure that's why they just start it, just fire it across the line in front of them, if you want to risk a false start
[Ruben:] Mm.
[John:] then fine.
[Ruben:] Yeah.
[John:] But use live ammo. [LAUGHTER]
[Ruben:] [LAUGHTER] Yeah []. [LAUGHTER]
[John:] Okay.
[Ruben:] Yeah.
[John:] So there's... there we are, and these are ten metres each time, so ten, twenty, thirty and let's say after... after a hundred metres... fifty forty. After a hundred metres you've got The Chair, that's where The Chair is. Oh not The Chair, the first fence.... The first fence.
[Ruben:] The man with the little flag's gone somewhere.
[John:] . Police notice saying no parking.
[Ruben:] [LAUGHTER]
[John:] So this time they get a good start, and they're off.
[Ruben:] Okay, so ten second intervals.
[John:] And let's say that they're
[Ruben:] Mm.
[John:] let's say they get up to full speed at erm after about... twenty seconds, it's alright, I'm just getting them
[Ruben:] Yeah.
[John:] to a gallop first.
[Ruben:] Oh r okay.
[John:] From a [LAUGHTER] racing start [].
[Ruben:] If that was a camera. [LAUGHTER] See.
[John:] [whipping sound]. N Not too much use of the whip.
[Ruben:] Yeah.
[John:] Right, so after twenty seconds they're up to full speed, and flat out, erm they're at forty five degrees. Okay? They've a speed on this graph it would be represented by an angle of forty five degrees. So they start off slowly,
[Ruben:] Right.
[John:] they get up to full speed, at twenty
[Ruben:] After twenty
[John:] seconds.
[Ruben:] After about there they get to full speed.
[John:] Yeah....
[Ruben:] So they start off... about there.
[John:] Mhm. Okay, and now that point.... If that's the point there... from that point on... the distance. They've reached full, they've reached that.
[Ruben:] Mm.
[John:] So if they kept on flat out, every ten seconds they'd go a further so many hundred metres, for ever and ever. They're going alright until they come to The Chair.... Now, so that's The Chair there. On the approach to The Chair, they slow down a bit.
[Ruben:] [whispering] []
[John:] Gather them up.
[Ruben:] [whispering] []....
[John:] And then just as they're on The Chair, they suddenly accelerate violently, to pop over it.
[Ruben:] As they
[John:] about a metre, a metre out from it when they take off. hop!... They pop up and over it, and when they land on the other side and they slow down a bit. Back to their, back to their normal speed again. And then the starter recalls them.
[Ruben:] [LAUGHTER]
[John:] Nice walk round The Chair, and down there. Are you happy with that?
[Ruben:] Yeah.
[John:] Okay. And let's have a look at this. Have you got a red pen? I sort of scrabbling about in my
[Ruben:] No sorry.
[John:] On the sa these, these graphs I think look better on the same axes, if you've got different coloured pens.... That one writes with its cap on. [LAUGHTER]
[Ruben:] [LAUGHTER] novelty. [LAUGHTER]
[John:] Now, how can you tell? That's distance, displacement, against time. Someone wants to look at, someone wants to look at that graph, and get some idea of the speed from it, the velocity. What would you tell them? How would you interpret that graph? I say well what's, what's happening here, what are they doing?
[Ruben:] For every ten seconds er... no for w for twenty seconds, the first twenty seconds, they're accelerating to twenty me metres per second per second.
[John:] Erm you accelerate to a speed. You can have an acceleration of so many metres per second per second. But you can't tell that from here, we're not, we're not, we're not into the acceleration yet. pinch that a minute. not to cut my nails. Ooh.... So there's the speed, just after they've
[Ruben:] Mm.
[John:] started.... Hardly moving.
[Ruben:] Yeah.
[John:] What's happening to the speed now?
[Ruben:] It's
[John:] And now, and now it reaches its peak. And what peak does it reach? Let's have a look.
[Ruben:] Highest.
[John:] Let's take from there to about, well there's four. About there to, and there's, there's two. So from that point to that point, you could work out the gradient. That's divided by twenty seconds. And what distance have they travelled, from there to there?
[Ruben:] Erm
[John:] From twenty to seventy... say seven call it seventy five. Twenty to seventy five.
[Ruben:] Fifty five. Fifty five.
[John:] Fifty five. They've done fifty five metres. Not very fast at all, this lot. This is not going to be a world record-breaking National.
[Ruben:] Bit like this year.
[John:] So they're going at fifty five metres every twenty seconds, and you can imagine Peter O'Sullivan, now look at this Pip, they're going at a, a nice steady fifty five metres every twenty seconds, what's that in metres per second? Well, divide that by twenty.... Almost three Yeah. Wow. [cough] [LAUGHTER]
[Ruben:] [LAUGHTER]
[John:] What's that in miles per hour? Well, that's almost one, one and a bit miles an hour that is.
[Ruben:] [LAUGHTER] Yeah.
[John:] Okay? So that's the, the, you can read off their speed from the graph, you can calculate it.
[Ruben:] Metres a second, sounds fast.
[John:] If we made those hundredths... then it might be a little. These were every hundred metres, four hundred metres is about a quarter of a mile, so quarter of a mile in twenty seconds. How far would you go in a minute?
[Ruben:] Four hundred metres in twenty seconds. So you'd go twelve hundred metres in one minute.
[John:] Twelve hundred metres, and that's three quarters of a mile or so, in a minute,
[Ruben:] So that means the race should be over in about
[John:] Ah.
[Ruben:] four mi in about four minutes.
[John:] About forty five miles an hour.
[Ruben:] [LAUGHTER] [LAUGHTER] That's []
[John:] So we've now gone from a real slow one to a really definitely record-breaking
[Ruben:] [LAUGHTER] Yeah []. Car speed.
[John:] So let's plot what the speed is at different points. What's the speed here? Well,
[Ruben:] Mm.
[John:] how does the tan of that, the slope of that compare with that? This bit here is twelve hundred metres per second we decided wasn't it? Er per minute, let's do it per
[Ruben:] Mm.
[John:] second, it's erm
[Ruben:] About three metres per second.
[John:] about three metres per second there. So that's three metres per second. This is, now let's, let's mark up here. Now here, metres per second. Right? Nought, one... two, three, and these are all metres per second.... From that point,
[Ruben:] Mm.
[John:] to that point, we're doing a steady three metres per second.
[Ruben:] Yeah....
[John:] So that's what, that's what the velocity looks like.
[Ruben:] Mm.
[John:] Steady. The distance is increasing steadily, the velocity is a steady velocity. And what's happening down here, the velocity has got less, here it's quite, quite a low velocity. The velocity is increasing, that angle is increasing. And that sort of, and it's increasing more rapidly at the beginning.
[Ruben:] Mm.
[John:] They're probably goes like this, running like this.... This is the velocity, and what happens here? Well the velocity gets a bit less,
[Ruben:] Yeah.
[John:] We're still trave we're still going further away as time goes on, but we're not going as quickly.
[Ruben:] As fast, yeah.
[John:] Now, acceleration,
[Ruben:] So as long as you're in the X positive, then you're moving in a positive direction, or a er
[John:] Depends on which graph you're looking at, if you're looking at your, your displacement your distance,
[Ruben:] Mm.
[John:] S against T,... right, then as long as you stay in the positive half, you are on the right side of the starting line, you haven't gone back and crossed over it. With the velocity, as long as it stays up there, it's travelling in the direction away from the starting line. When it comes back over the starting line
[Ruben:] Yeah.
[John:] then it's travelling in the opposite direction, it is now
[Ruben:] You could have gone
[John:] velocity in the opposite direction.
[Ruben:] Yeah.
[John:] And if this goes round a sort of roughly circular track, the we're not measuring velocity, we're measuring speed. So acceleration then.... What is acceleration? Well the acceleration is the steepness of this graph. How quickly the speed is increasing as time goes on. So the acceleration here is sort of pretty sharp acceleration to start with, then a fairly steadyish acceleration,
[Ruben:] Mm.
[John:] yeah? At so many metres per second per second. And then, what happens there, along this bit? What's the acceleration doing from there
[Ruben:] Remaining constant.
[John:] to there. The velocity is remaining constant.
[Ruben:] The s s s
[John:] The acceleration is, how is the velocity changing, how much is it changing?
[Ruben:] Three metres
[John:] It's constant, three metres per second, constant velocity.
[Ruben:] Yeah.
[John:] Got it? So the police got his radar gun on your horse,
[Ruben:] [LAUGHTER]
[John:] [LAUGHTER] [LAUGHTER] you don't get there first [].
[Ruben:] Yeah.
[John:] Right, what's he doing? He's doing thirty mile an hour. What's he doing now? Thirty miles an hour. What's he doing? Thirty miles an hour.
[Ruben:] Mm.
[John:] What's his acceleration?
[Ruben:] None.
[John:] . Good. Brilliant. This car does thirty miles an hour to thirty miles an hour in fifteen years.
[Ruben:] [LAUGHTER]
[John:] [LAUGHTER] Right []? It doesn't matter how long it takes, it's going to, you know it's not accelerating. So its acceleration is this.
[Ruben:] Mm.
[John:] a green one or something, but I'll do it as a red one
[Ruben:] That's pens for you.
[John:] So that's its acceleration
[Ruben:] Mm.
[John:] over that. So just over this short stretch here, which is the simplest part of the curve, a nice steady velocity, a nice steady speed, a fixed three metres per second, forty five miles an hour, whatever it is.... That's its speed. So that's, that's, sorry, that's its, how its distance is changing as time goes on. This is over the same event that we've recorded... how the speed is changing, it isn't,
[Ruben:] Mm.
[John:] it's fixed, and this is how it's accelerating, well its acceleration zero all the time, he's not getting any faster. but how can his acceleration be zero? He's still doing the nice steady thirty. Yes, but he was doing that a second ago. And we're doing it in a second time. No acceleration, no faster no slower. So those three graphs for the simplest bit, superimposed. If you have a look through your, that really good book,
[Ruben:] Mm.
[John:] erm, you don't like looking through it, but can I just look through for you and learn it for you.
[Ruben:] [sneeze]
[John:] No. If, if you can look through that now, you'll get a lot of benefit
[Ruben:] Yeah.
[John:] out of what we've been doing with this. I mean don't n necessarily mean this evening, or fairly soon. Erm... cover everything else up, and just concentrate on one. And start with the how does the distance change as time goes on? How far from the start line have they got?. And look at that, interpret it, think oh well, the distance is increasing only fairly gradually as time goes on, but when it gets to there, the distance is increasing at a constant rate, as time goes on, so we've got a constant speed. But now it's not increasing as quickly, its gradient has come down a bit. Its gradient is the speed all the way through here. So then from that graph, try and predict before you look at it in the next picture
[Ruben:] Mm.
[John:] how the, how the velocity itself is changing. I mean well you've, you've looked at what you think acce what the, the velocity is, which is the gradient on the distance time graph. Then check it against their velocity time graphs to see if it's correct. You get some funny bits or going the wrong way. Try and and work out why. And then when you've got, when you look at their velocity, velocity against time, try and work out what the acceleration is again it's the slope.... when it's not changing. you're looking at the velocity, that'll be velocity, so where it's not changing, there, the slope is zero, no acceleration. Now if you take that through the examples they give, cos they're quite good ones in there, it will mean a lot.... Erm I don't think, don't think there's a year since what? Since what? Well since that fracas with the Grand National that they've er that they've missed this. They always have a distance, time, velocity acceleration one of those on it. Probably only distance and time, distance and velocity. So probably won't need to worry about the acceleration ones. Very important for the physics, because what's happening at that point as far as the physics is concerned, when the acceleration is zero, not, think of a racing car accelerating up to top speed, down the straight and it just can't go any faster, it's got up to a hundred and eighty miles an hour, and it's
[Ruben:] Like going on a flat road, without putting the brake
[John:] Yeah.
[Ruben:] the gas.
[John:] Flat road, no brakes, and he's got his foot hard to the floor, it won't go any faster. Erm what's the, what are the physics of it?
[Ruben:] The physics?
[John:] Or the kinetics or. Without looking it up. What, what's happening? Why won't the car go any faster? Usually you put your foot down hard, it goes faster. Why won't that car go any faster?
[Ruben:] Because all the power its
[John:] Putting all its power into it, okay.
[Ruben:] Erm... its engine
[John:] What's stopping it?
[Ruben:] Engine's reached its
[John:] No,
[Ruben:] top force
[John:] engine could go faster. Top force. Okay the engine is pushing as hard it can to try and, like a lot of people running behind leaning on this car pushing it forward, and why, why isn't it going f usually if you keep pushing something it goes faster and faster.... Until
[Ruben:] Mm.
[John:] it reaches top speed. Why does it reach this top speed?
[Ruben:] Erm... Is it the backward force? The, because it's being pushed back, cos the opposite force, when you push, something else always pushes back with erm... if you push on this table,
[John:] Pushed back on me, I felt it, then shot off. And it stopped when it hit something. It didn't have much friction, so it was going along there alright. Erm... Why do they do wind tunnel tests on cars?
[Ruben:] A wind tunnel?
[John:] Erm
[Ruben:] [LAUGHTER]
[John:] a wind tunnel is where they put this, put the car in stands and they blow air at it, and they little smoke trails that show how the air travels over the car. Where it breaks up and makes eddies, turbulence.... Right. It's the wind, it's, it's, try erm, ever tried running through fairly shallow water? To about just over your knees?
[Ruben:] Pretty difficult, yeah.
[John:] It's the drag of the water, dragging you back. Well when you walk in air, there is a very slight drag, but you don't feel it. Have you tried cycling into a strong wind?
[Ruben:] Yeah.
[John:] Especially recently?
[Ruben:] Yeah, yeah.
[John:] And that camper is, is quite big on top,
[Ruben:] A lot of wind pushing
[John:] I went down yesterday to erm.... And it was getting dragged back and the engine was going like mad trying to push it along and the wind was pushing me back, and I was thinking am I going forwards or backward here?
[Ruben:] [LAUGHTER]
[John:] Erm normally, erm, normally it wouldn't, it wouldn't do ninety,
[Ruben:] [LAUGHTER]
[John:] but er it was all it could do to do sixty.
[Ruben:] Yeah.
[John:] was the wind.
[Ruben:] I suppose, if the wind was on its back, then it'd get pushed along,.
[John:] It does, yeah, on the way back on the way back it was pushing it a bit. Because of the way the drag works, it's better if it it's worst if it's not quite head on, but slightly offset from head on. Similarly with a tailwind, it's not exactly behind you, but just a bit to one side. You could feel
[Ruben:] Is that they're so low to the ground?
[John:] That's why all cars a few years ago started looking the same, now they all look sort of like this. R rounded, rounded there and not much there, and sawn-off at the back.... And they're low to the ground, so the ground effect and that so the air can go over there,... breaks up here and gets thrown off... erm... it's just the air. The