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Two Bicycles

Patrick Ford
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Hey, guys, let's check out this example. Problem. Work it out together. I've got a position time graph that is shown over here. Except now. It's not for just one object. I actually two moving objects or to moving bicycles labeled A and B. Now, the first question is asking us at what time or times do the bicycles have the same position? So what does that actually mean on this diagram here on this graph? Well, if I'm looking at a position Time Grafton from our table are of conceptual points in position graphs, then we know we're just going to be looking at the values. So what this question is really asking us for Is that what time or times on this time access over here are the bicycles at the same exact value. So let's take a look here. Well, bicycle B is gonna be at this value over here, and then bicycle is gonna have this value over here later on. What happens is when the two lines will cross. This is one of the points where they have the same value, so t equals one is one of our times. Let's just see if it happens again and later on in the diagram we'll be or sorry, bicycle A. We'll just continue on like this and the values over here whereas bicycle be the values, they're gonna be over here. Notice how these are not the same. But eventually what happens is B is gonna catch up to a or the lines are gonna meet again right over here. Here is where the values are also going to be the same. So these two points here correspond to the same values. And so therefore there at the bicycles are at the same position. So it's t equals one and T equals four seconds. Those there are two times. There's never anywhere else on the diagram in which these two lines will cross each other. So let's we want to be now. What time or times did the bicycles have Roughly the same velocity? Well, on a position time diagram. Remember? Now we're looking for the velocity, which means we're not looking for the values were looking for the slope. So what this question is really saying is, where do the bicycles or what point to the bicycles on this time diagram. Do they have the same slope, not the same values. It's not going to be here or here. They don't have the same slopes here, so let's check it out. Well, the slope for a is actually just a straight line. So in other words, it's always going to be this line over here. It never changes its constant velocity, whereas for be what happens is it's a curvy position graph. So we know that there's gonna be some acceleration and the velocities air constantly gonna be changing. So where do these two lines have the same velocity? Well, I could basically trace out with the instantaneous velocity. Looks like here it's kind of hard to visualize it or c it really clearly. So this is like the instantaneous velocity. And then right around here starts to trend upwards like this. And I'm basically looking for where it matches this approximate sort of steepness here and right around at three. If I were to draw the tangent line, the tangent line looks like this. So let me erase all the other ones, so it just looks a little bit clearer right here. So at this point right here, the velocity of B is this line on the velocity of a is this line. So that means that that is actually the point at which they have the same velocity. So let me go and actually write this in blue just so you can see it's super super clearly so the same slope is going to be right here at this point over here. So that means that the time is T equals three. Then what happens is the slope is gonna continuously increased for be and it will never be the same as a again. So that means that there's only one time that this happens. It's a T equals three seconds right over here. So notice how these two slopes of the same Alright, guys, um, that's it for this one. Let me know if you guys have any questions.