>> Hello class, Professor Anderson here. Let's talk about relative motion. This is a very important concept in physics, and it's this idea that motion is all really relative, okay. So, let's say we do the following experiment. Okay, I have a pen in my hand here, and as I walk across the screen and I toss that pen straight up and down, what do you guys see in the audience? Somebody have a thought? Let me know. What do you think? What's the curve that you observe? Yes, Laura? >> A parabola. >> A parabola. Right? Parabolic motion. So, it would look something like that. Like you said, it is a parabola. Okay, once that object leaves my hand, it is, of course, in freefall. The only thing acting on it is gravity, and we know that projectile motion maps out a parabola. But, this is what you observe. This is what it looks like to you. But, what does it look like to me? Alright, as I walk across and I throw that object straight up and straight down, what I see is something very different. [ Silence ] To me, it looks like that thing went straight up, and straight down. My coordinate system was moving with me. Your coordinate system was stationary in the lab frame. Okay, and so those are two different coordinate systems. And, in fact what we should do is probably call this the x prime y prime frame to denote that it's different than the x y frame. And so, this idea of relative motion is really important, and the person that came up with the idea originally was Galileo, and he wrote down something called a Galilean Velocity Transformation. [ Silence ] And, we're going to see how that works. Okay, let's say we have two different observers. That could be like me, and you, and S is the stationary observer, that's you. S prime is the moving observer, that's me. Okay, and I am moving at a speed V naught. Alright, how can we make sense of what you observed versus what I observed? Well, the way we do it is the following. Let's draw an origin for the S frame, and let's draw a different origin for the S prime frame. Okay, and we know that the S prime frame is moving along relative to the S frame, and it's, in fact, moving at speed V naught, so that at time t it will traverse a distance V naught t. Okay, just increases as a function of time. Now, something that we observe is sitting up there at A. That could be the position of the pen when I toss it. O, to A is what you observe. That is the position vector that you observe, and we can call that r. But, O prime to A is what the moving observes and that we can call r prime. Alright, so this, maybe looks a little complicated, but we are familiar with vectors, right? We know how to add up vectors. It looks like if I take this vector, V naught t and I add r prime I'm going to get the sum of the two which is r, and that's exactly what we did. r is just the sum of V naught t plus r prime. Okay, and now you can rewrite this for r prime, if you're interested. r prime equals r minus V naught t. And now, you can solve for V, right? What is the velocity observed in the moving frame? It's just the derivative of r prime with respect to t, and so I need to take a derivative of this thing. What do I get? I get d r, d t minus V naught. But, d r, d t is just the speed observed in the stationary frame. And so, this is the Galilean Velocity Transformation. What is the velocity observed in the moving frame? It is the velocity observed in the stationary frame minus the velocity of the moving frame. You can, of course, rewrite this. The velocity in the stationary frame is going to be the velocity in the moving frame plus that. Okay, so these are known as the Galilean Velocity Transformations, and it should make sense to you, right? You just add or subtract velocities. But, you have to treat them as vectors, okay? Let's try an example of this.