Hi and welcome to module seven of three dimensional dynamics. So, we're moving right along in the course. Thus far, we're studying kinematics of three dimensional motion, and we've looked at angular velocity and angular acceleration. And today we're gonna start the next topic of velocities and accelerations in moving reference frames. And so our learning outcome for today is to derive the equations for velocity expressed in moving frames of reference. And so here's our situation. As you recall from my earlier course in two dimensional dynamics, I'm a hang gliding pilot, as you can see here in the video. And when I did the two dimensional dynamics course, we restricted the motion to just planar or two dimensional motion, whereas now we know that in reality this is obviously three dimensional motion. So, we're gonna go and develop the expressions for three dimensional motions. I've got my Frame F, which I'm gonna call my fixed frame. Although, as you recall from the past, it doesn't necessarily need to be fixed in an inertial reference frame. Here, we're calling F the Earth with some person observing me flying around in the hang glider. And then I have a moving Frame B that is attached to a car that may be moving along the ground, and also observing the hang glider. And we wanna express the velocity in those two frames. And so, the first of all, the velocity of the hang glider in Frame F is just the derivative of the position vector in Frame F from point O to point P. The velocity of the hang glider in the moving frame, or Frame B, is just the derivative of the position vector of r O' to P. And then finally, the velocity of the car is the position vector derivative from point O to point O'. So, here's our situation again. Let's begin by writing a vector addition equation for the position vectors. And so what I have here is that the position vector from O to point P is equal to the position vector from point O to point O' plus the position vector from point O' to P. And so my question to you is, we wanna find velocities that expressed in moving frames of reference, and so what should we do next? And what you should say is we're gonna need to differentiate in Frame F. Okay, so we're gonna differentiate everything in Frame F, I'm gonna start now. And I'll have the derivative of the position vector from O to P in the F frame is equal to the derivative of the position vector from point O to O' in the F frame, plus the derivative of the position vector from point O' to P in the F frame. And so we'll go the next step and we see that the position vector derivative from O to P from the Frame F is just going to be the velocity of point P with respect to the F frame. So, we can write this as the velocity of point P with respect to the F frame equals. Now this term is the derivative of the position vector from point O to O' in the F frame, and so that's just going to be the velocity of point O', or sometimes I call it the absolute velocity of O' with respect to the F frame. So, we'll write that as the velocity of O' with respect to the F frame. And then I'm gonna have to be more careful about this next term, and I'll talk about that on the next slide. So, let's go on from there. And so here's where we left off. Now, in taking the derivative of the vector r from O' to P, the position vector r from O' to P in the F frame, I have to be careful because r from O' to P is expressed in the B frame, but I wanna take the derivative in the F frame. And so my question to you is, how should we go about doing that? And go ahead and try to do it on your own, and then come back and we'll do it together. And so what you should say is, again, we're going to use the derivative formula, and the arbitrary vector A that we're going to choose this time is the vector expressed in the moving frame, which is r from O' to P. And so we're gonna let A equal r from O' to P, and what I get is, now I'm working on this last term. Whoop, I'm working on this last term. And so, we get r from O' to P dot, in the F frame, that's this term, is equal to, now it's this term, we've got r from O' to P derivative in the B frame, and then I have how the B frame is moving with respect to the F frame. So, that's the angular velocity of B with respect to F crossed with the vector A itself, which is r from O' to P. And so this equals now, what's another way we can express r from the derivative of the position vector from O' to P taken in the B frame? And what you should say is okay, from the B frame looking at the time derivative of that position vector, that's just the relative velocity of the point P with respect to the B frame. And so that's V of P with respect now to the B frame. And we still have omega B with respect to F, crossed with r O' to P. So, this is what we just found. All we have to do now, is we have to substitute this in here, and we'll come up with our expression for velocities expressed in moving frames of reference. And so now I have the velocity of P with respect to F is equal to the velocity of O' with respect to F, so that's that term. And then I've got plus the velocity of P with respect to B plus omega B with respect to F crossed with r from O' to P. And so that's our expression for velocities expressed in a moving frame. What I'd like to do now is I'd like to just write out in words what each of those terms are, so we make sure we have those cemented in our minds. So we've got, this is the absolute velocity of P in the F frame, so this is the absolute velocity of P in Frame F. And then, excuse me, this is the velocity of the origin of the moving frame, O' with respect to F. And so that's the absolute velocity of the origin of the moving frame. Let's change colors here. So we've got absolute velocity of origin of moving frame. And then this term, as we said before, is the velocity of P as seen from the moving frame. So, that's the relative velocity with respect to the moving frame. So that's the relative, let's change colors again here. Relative velocity now of P in frame B. This next term is just the angular velocity of the moving frame. And then finally, the last term is the position vector as expressed in the moving frame. Okay, so that's the equation for velocities expressed in moving frames of reference. We can actually write this in a little bit of a shorthand, which will help as we solve more problems, kinda cut down on the complications here. So, what we can say is the absolute velocity of P and F, we'll just call the velocity of point P, the absolute velocity of point P. We'll call this vector, this is the vector r from O' to P, little r. We'll call the vector that's locating the origin of the big reference frame, R. And so the absolute velocity of the origin of the moving frame is just equal to R dot, so that's that expression. Then, since this is the relative velocity of P in B, we'll just call that VREL. And since we're working with the moving frame here, I'll just say plus omega, knowing that that's the moving frame's angular velocity crossed with, since we always express the position vector in the moving frame, we'll just call that r. And so that's a real simple version of the equation, kind of a shorthand. And if you go back to my earlier course in two dimensional dynamics, you'll see that, that's the same expression we came up with there, except that the angular velocity, this was planar motion then, and the angular velocity only had k components. In this case, each of these vectors can now have three dimensional components. And so we'll go forward, and actually solve some problems in the next module.