Hi. This is module 24 of two dimensional dynamics. Today's learning outcome is to derive the equation for the velocity of the same point relative to two different frames or bodies in planer motion. And, so, here's the situation. You'll saw, you saw in the introductory click, clip that I'm a hang glider pilot. Now, obviously a hang glider moves in three dimensions, but we're going to restrict our our motion in this case to just looking at planar motion. In my later course, Advanced Engineering Dynamics, we'll go ahead and look at 3D, motion as well. But here I'm going to call my F frame or my fixed frame the Earth. And then I'm going to weld a moving frame to my hang glider, which I'm going to call frame B, and then we're going to look at the motion of a common point, maybe it's a bird, from the frame B, or from frame F. And remember, frame F is, I call it fixed but it doesn't necessarily have to be an inertial frame, maybe we would attach frame F to a car, rolling along the ground. And frame B to a hang glider. When I say fixed, it's just fixed in respect to the other frame which I call moving. And, as I again said last module, you can, have several modules. You, you or several frames. You may go to hang-glider, back to a car, back to an Earth. You can link these all together. So let's start by writing the position of vector here vectorally. And so we're going to have the, position from O to P from the origin of the F frame to point P equals the position from O to O-prime. Which is the origin of the F frame to the origin of the B frame. Plus r from O-prime to P. And that's a, a good vector addition. Or vector equation for the positions. Now to find velocities, what do I need to do? And what you should say is just differentiate and so we will go ahead and differentiate here. And we've got now the velocity of P and this velocity of P, we take this derivative is going to be with respect to the F frame equals the velocity of O prime with respect to the F frame. Plus, we need to be careful about this derivative. This is the derivative of R O prime to P. Which is a vector expressed in, what I'm calling my moving frame, and I want to take that derivative with respect to the F frame. And so that derivative is a little bit more complicated. So I'll write it like this. And my question to you is, how are we going to take that derivative and, and, and take it properly in the F frame? And the answer is, you should use that derivative formula, that we derived in the last module and I said was real important. So, for this third term, the arbitrary A vector, which we'll, be taking the derivative of this time, is actually going to be dire, the position vector from O prime to P. This vector. And so I've got, V of P, with respect to F. Is equal to, V of O prime, with respect F, plus now we want, the derivative of R O prime with respect to P in the F frame, so that's going to be the derivative of R O prime the P in the B frame. Plus, how the B frame, the angle of velocity of the B frame with respect to frame F, so that's going to be theta dot K for planar motion, two-dimensional motion, cross with a vector itself expressed in the moving frame which is R from O prime to P. And so that's, that's how we take that derivative. So let's pick up from there. That's the expression I just wrote. I can actually, this, this is, this is actually the equation for velocities of the same point relative to two different frames or bodies. I want to explain each term. And I want to use a little bit of a short-hand here. I'm going to call the velocity of P with respect to F, the absolute velocity in F. Okay? So, this is going to be just the velocity of P and I'm going to call that the opps, absolute velocity in frame F. And then. That's going to equal, the velocity of the origin of the moving frame with respect to F, I'm going to call the absolute velocity of O prime, with respect to F. So this is, and I'll just call that the velocity of O prime. And, this is the, absolute velocity of the origin of moving frame B in frame F, and then I've got plus V P with respect to B. Now, that's the velocity of P with respect to my moving frame. And so I'm going to term that the velocity relative. So this is going to be velocity and I'll just abbreviate REL. So this is the relative velocity in moving frame B. Relative velocity of our point in moving frame B. And then I've got the angular velocity of frame B with respect to frame F. So plus, and that's going to, I'll just use the term omega as a vector. That's the, angular velocity of, frame B. With respect to frame F. And then I'm going to cross that with r, and r is always going to be expressed in the moving frame, so that's just the position in the moving frame B. So there you have it. That is now the derivation of the velocity of the same point relative to two different frames or bodies. And we'll use that derivation to solve some problems in the next module.