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 have two moving objects or two 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 graph, then from our table of conceptual points and 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 at what time or times on this time axis over here are the bicycles at the same exact value? So let's take a look here. Well, bicycle B is going to be at this value over here, and then bicycle A is going to 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=1 is one of our times. Let's just see if it happens again later on in the diagram. Well, bicycle A will just go on like this and the values are over here. Whereas bicycle B, the values are going to be over here. Notice how these are not the same. But eventually, what happens is B is going to catch up to A or the lines are going to 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, therefore, the bicycles are at the same position. So it's t=1 and t=4seconds. Those are our two times. There's never anywhere else on the diagram in which these two lines will cross each other. So let's move on to B.
Now at what time or times do 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, we're looking for the slope. So what this question is really asking is where do the bicycles, or here. They don't have the same slopes here. So let's check it out. 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. It's constant velocity. Whereas for B, what happens is it's a curvy position graph, so we know that there's going to be some acceleration and the velocities are going to be constantly changing. So where do these two lines have the same velocity? Well, I can basically trace out what the instantaneous velocity looks like here. It's kind of hard to visualize it, or see it really clearly. So this is like the instantaneous velocity. And then right around here, it starts to trend upwards like this. And I'm basically looking for where it matches this, you know, approximate sort of steepness here. And right around at 3, 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 and 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 ahead and actually write this, in blue, just so you can see it 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=3. Then what happens is the slope is going to continuously increase for B, and it'll never be the same as A again. So that means that there's only one time that this happens. It's at t=3seconds right over here. So notice how these two slopes are the same. Alright, guys. That's it for this one. Let me know if you guys have any question.