Onto part three of our analysis of the Michelson Moorely experiment. Remember we did the hail wind tail wind case using the airplane example. And found that the time-round trip time was of this form here. 2D over Vp, where V sub-p is the plane, and then times a 1 over 1 minus Vw squared Vp squared factor where Vw's squared. Vw is the velocity of the wind there. So now let's do the plane situation but with a crosswind this time. So instead of going up and down, the wind is still coming down here or across, and we're going from A to B with our plane. So velocity of the plane from A to B and then back again. Distance D. Again, we want to analyze what is the time. That it takes to get there. And to understand this situation, we actually gotta use another analogy that's maybe even more intuitive for us just to see what's going to happen here with the plane. And then, of course, whatever we analyze here will apply to the situation of a light beam traveling from the supposed ether because it's just a way of traveling through a medium. And if the medium is moving, if we've got that ether wind or ether breeze going, whether it's headwind/tailwind type of ether breeze to the light beam or a crosswind to the light beam. So the light beam's here and maybe the ether is It's coming this way depending on how we're moving or whatever. Then the analysis is still going to apply. So velocity of the plane here going back and forth. Cross wind, what's analogy we want to use here just to more intuitive idea of this. And that is think about swimming across a river, okay. So if you've got here is the river. Here's the current flowing in the river. And let's say I'm at point A here and I want to swim directly across to point B. And we'll say there's a distance D from A to B directly across. Well, we know that if I were to swim straight like that, the current of the water is going to push me, downstream. So if I go, if I try to travel straight across, I'm actually going to end up, going like that. And I will not end up at point B, I'll end up someplace downstream. Okay, so if I actually want to end up at point B, what I have to do is swim up like this, in that direction, and then the current will be pushing me down. In fact it will push me down that much right there, assuming I've more or less drawing the arrows the same there. And for those of you who know the fancy word for this it's a little bit of vector analysis. But you just add the arrows together, head to tail. So, here's one arrow, that's my actual velocity here going across. And the river is pushing me down with this velocity here and therefore my resultant motion through the water, which is what I'm interested in, is going to be straight across. More or less there, I'm going to get to point B instead of ending at balance. I have to swim essentially partly upstream so the current carries me downstream so I end up at B here. And, that's exactly what the analysis is here with the plane is that instead of the water current here, it's the wind that's pushing the plane off course, so that if it tried to just go straight across in a straight heading from A to B, it'd end up off course down here. So instead it has to head into the wind. At a certain angle which we don't have to worry about fortunately, but had into the wind at an angle such that it ends up at B where it's supposed to be. And so, what are the quantities here? Well, it's flying through the air at velocity, V of P, so that's the actual. Velocity of the plane through the wind or if you want the swimmer. The velocity, maybe we're swimming, the velocity of us through the water there. And then the current is V sub W. In our plane example the velocity of the wind. And so that's going to be this quantity right here this side. Notice we have the triangle here and so that's going to be V sub w there. My arrows are a little messy but that's the arrow for the wind, that's arrow for velocity of the plane. And then what's this arrow here, what's the actual velocity from A to B that we're travelling, we'll just call that V actual. [NOISE] Okay, so, if I want to end up at B, I have to, point my plane in a direction away from B at a certain angle, so that the wind carries me such that I end up at B. But, obviously I have to use a little more fuel this way, because I'm traveling actually a farther distance. There, in a sense, I have to have a higher velocity to end up over b with a velocity actual. So what is v actual? We know what v sub p, the plane, assuming we're given that, I mean 300 kilometers an hour or whatever we happen to be using. 500, 600, so VW is the velocity of the wind, we're assuming that's a given, and so what's V actual here? Well, here's something I don't think I included in the math review, but we'll put it in right here. Remember the Pythagorean Theorem, I essentially have a right triangle here. I've got something that looks like this. This is a right angle. I've got V actual here and velocity of the plane there and velocity of the wind there. I'm given Vp and Vw, those are quantities I'm working with here. What is V actual? Because, note, that's what I want. V actual here is going to tell me how long it takes to get from A to B and then back again. And I know in this case it's nice because it's symmetric. In other words, A to B, I've got the crosswind. B to A I've got the same crosswinds. If I do it for one case A to B, then the other case, B to A will be exactly the same. So the total time would just be twice the one leg time. And the velocity. From A to B here will be V actual. That's the net velocity from A to B straight across there, even though the velocity of the plane has to be up like that because the velocity of wind, of course, is pushing it down. So, what is V actual over here? Well, Pythagorean theorem tells me that this squared plus this squared equals this squared. Remember? So if I want this, I can write v actual, Squared equals the hypotenuse here squared, v p squared minus V w squared or, more precisely if I just want to do v actual equals the square root of v p squared minus v w squared assuming that we're dealing with positive quantities here for the velocity, of course. Okay, so that gives me the net velocity of the plain from a to b or the swimmer from a to b if you want think about a swimmer. And so therefore I can do the time now so let us erase our river here. The plane situation like that, we know what V actual is now, we just found that out. So, what's the time? So, let's just go first of all, A to B. A to B travel time is going to be the distance, so time. Taken is going to be the distance simply divided by the velocity. The net velocity in this case which is v actual. And v actual we just discovered was square root of vp minus vw squares. So this becomes D divided by square root of vp squared minus vw squared. And that's from A to B. From B to A is exact, the same analysis because we still have a cross wind and it's at the same orientation. So nothing really is different. It's just the other side of the plane in this case, but you still have to arrow up into the wind, and so we can say a to b and then b to a is just going to be that again. So I get twice that for the total time, so we'll say, squeeze this in here, total time is going to be 2 d times that. Okay? Now we're going to rewrite it one more thing here, one more time here, one more step. So, let's do our little mathematical technique here and we're going to factor out a v p squared from here. We'll do it step by step so you can see what we're doing. 2D over square root of VP squared times 1 minus VW squared over VP squared. That was a little technique, trick. We reviewed earlier, reminded earlier, in the earlier video clip part one, this, you factor out of vp squared from each term. So it's vp squared times one, vp squared times that, so that's equivalent to that. I can then split this into two square roots like we did before, square root here, square root here. This is two d over the square root of. Too much there, square root of vp squared times the square root of 1- vw squared over vp squared. And of course the square root of vp squared is simply vp. So this becomes 2D all over v p and I've got this other factor here which I'll write as 1 over the square root of 1 minus v w squared, v p squared. Okay. See where we got that? So here I had the 2 d, this just becomes the v p and the second here is the one over this I just pulled out as a separate term there. Okay, so let's just rewrite that, clean it up a little bit here. So the cross wind case, because that's what we've been working on here is time. Equals (2 D over v p) times, 1 over the square root 1- VW squared over VP squared, and note something important here. Something very interesting, this was the crosswind case, we earlier did the headwind/tailwind case. They're slightly different. They both have the 2 d over v p factor in there, but note, in this case, it's 1 over 1 minus v w squared over v p squared. Here, same thing but the square root. Slightly different. So, in part four of this, we'll see how we put this together and how Michelson Morley put this together. And see how it applies actually to the luminiferous ether and light waves travelling through that and how they did their experiment and what the result was of that experiment.