>> Okay, so Galileo said the following, "You can add or subtract velocities, but you have to be careful and do it as vectors". And so, anytime we're dealing with Galilean Velocity Transformation, we're always talking about frames that are moving along, coordinates that are moving along at constant velocity, no acceleration in those frames, okay? Constant velocity motion between the two frames. So, let's try the following example. Let's say that we are standing on a train, and now we are walking inside that train, okay? And, you probably already know the answer to this, but if the train is moving at 30 miles per hour, and I'm walking inside it at 10 miles per hour, what's the velocity of me, relative to the Earth? Anybody know what that is? If you know, raise your hand. Yes, Samantha? >> Samantha: 40 miles per hour? >> 40 miles per hour, that sounds good, everybody else agree with that? 40 miles per hour? Is there any other possible solutions? Yes, Sean. >> Sean: It could be 20 miles per hour, for walking the other direction. >> It could be 20 miles per hour, absolutely, if I'm walking with the train in motion, it would be 40 miles per hour, if I'm walking against it, it would be 20 miles per hour. Any other possible solution? Yes, Doug. >> Doug: It could be the same if you're walking sideways. >> Okay, it could be something else if you're walking sideways. Maybe not the same as the train or the same as you, but something else. Right? So, the idea is that we're going to add-up vectors, okay? And, let's identify what those vectors are. So, we have the vector for the velocity of the train relative to the Earth. And, this is train relative to Earth. And so [inaudible] the velocity of the train relative to Earth? And now, we have a velocity of the person. Relative to the train. And so, VPT is a Velocity of a Person. Relative to the train. Okay. But, what we really want to know is, "What does somebody out here, observe?" Okay, this is the observer in the stationary frame. And, they're going to say, "Oh, the velocity of that person, relative to the Earth, is what?" And, what is the velocity of a person relative to the Earth? Okay, so, what we said was, "If the train's moving at 30 miles per hour, and you're walking at 10 miles per hour, then we could be moving at 40 miles per hour, relative to the Earth. But, we could be walking the other way, and then that would be 30 minus 10, which would be 20 miles per hour, relative to the Earth". So, let's put the bounds on it like that. Let's say that the speed of the train, relative to the Earth is, what we've said? 30 miles per hour. 30 MPH, and the speed of the person on the train is 10 miles per hour. [Inaudible] maybe faster than a walk, more like a run. And now, we need to figure out VPE. What is the biggest that VPE could be? What's the biggest the VPE could possibly be? We already said the answer to this, alright? That's if you're walking in the same direction as the train moving. The biggest it could be is 40. What about the smallest it could be? Anybody has a thought on that? Yes, Ian? >> Ian: 20 miles per hour. >> 20 miles per hour. Correct. Those are the bounds on VPE. This already makes sense to you, but, let's put it in the context of vectors. Okay, we said that the train is doing this -- this is V of the train, relative to the Earth. If I add to that, the vector of the person, relative to the train, then I just do the tip-to-tail method and, I get a velocity of the person, relative to the Earth. Okay, but what about when I go the other way? If I'm going the other way, then the velocity of the train, relative to the Earth, I have to add a negative velocity of the person, relative to the train. And, therefore the sum is going to be that, little bit, right there. Okay? So, in that case, we're going to get 40 miles per hour, and in this case, we're going to get 20 miles per hour. But, of course, we don't have to add those vectors in the same direction. You could be walking sideways, relative to the train. And so, this would be the person relative to the train, and therefore, the sum of those two is right there. This is the person relative to the Earth. And, in this case, if this is the right angle, now we can solve this, VPE squared equals VTE squared plus VPT squared. And, what did we say it was? The train, relative to the Earth, was 30 miles per hour. A person relative to the train was 10 miles per hour. So, we get 900 plus 100, which is 1,000. And so, the VPE equals square root of a 1,000. And, we punched this in earlier, and what do we get? We got 31, I think? 31.6 miles per hour. So, look at this picture right here, If I take VPT, and I move it in a different direction, I can either add it to VTE, I can subtract it from VTE, or I can go somewhere in between, okay? And so, the VPE is going to go from this short arrow, to a maximum of this long arrow, or something in between, too. All right, questions about? That one? Is that relatively clear? Pun intended. All right, if not, definitely come see me in office hours. Cheers.