Write a short description of the motion of a real object for which FIGURE EX1.20 would be a realistic position-versus-time graph.

Video transcript

Hey, everyone in this problem, we're asked to draw a graph of position versus time for the given situation. OK. So the situation we have is a cyclist starts cycling from his house and rides at a velocity of 20 MPH for the first three hours, he moves at a velocity of 25 MPH for the next two hours. And then he continues at a speed of 15 MPH until he reaches his destination four hours later. OK. So we're given four answer choices here. A through D, all of them show position versus time curve. OK? We have time and hours on the X axis and X and miles on the y axis. OK. So we're gonna come back to these as we work through this problem. But let's think about what this position versus time curve is gonna look like we're given velocity. So the first thing is to realize how we can relate velocity to position. OK. So let's recall that the velocity it's going to be equal to the slope of our position time Kirk. OK. So the velocity is the derivative of that position time curve which when we're talking about a graph means that it's the slope. All right. So we know that our velocity is gonna be this slow. We have a positive constant velocity for each segment. So if we have a positive constant velocity for each section, that means that we should have a line. OK. The line coming from that constant velocity, it should have a positive slope and we need this positive slope through. All right. Now, we also know that if we have a larger velocity, we have a larger slope which tells us that we have a steeper line. Ok. So the middle sector, middle section where we're going 25 MPH, that fastest speed should be the steepest, right? All right. So based off of this information alone, we can already kind of narrow it down in choice. A from zero hours to three hours, we have this negative slope in choice B we have a negative slope from five hours to nine hours and same with option C, we have a negative slope from five hours to nine hours. So it's already looking like option D is gonna be the only possible, correct answer based on just the sign of the slope alone. But let's do a little bit more in terms of the calculations just to verify that everything does indeed match up. So let's look at the distance traveled in each section. Hm All right. So for the first section, we're gonna call it section one. And let's go up to the question. We're gonna do section one in blue and this is where the cyclist starts from his hos rides at 20 MPH for three hours. All right. So, recall that the velocity or the speed, does it go to the distance traveled? Divided by the time, which tells us that the distance traveled is gonna be b multiplied by T. Now, in this case, the speed V is 20 MPH multiplied by the time three hours. Ok. At the speed, the unit of hours is gonna divide it. We're gonna be left with just miles and we have a distance travel of 60 miles. All right. What about section two? Get speed is distance divided by time. So the distance we're interested in is the speed multiplied by the time going back up to the problem. Ok. In this second section, we have a speed of 25 MPH or two hours. Ok. So back to our calculation substituting in those values 25 MPH multiplied by two hours going to give us 50 miles traveled in this second section. And finally, for section three. Mhm. Again, the distance is gonna be the speed multiplied by the time we go back up to look at the details. Hey, we have a speed of 15 MPH for four hours. We multiply the two together to give us a distance traveled of 60 miles again. OK. So now that we have this information about distance, let's go back to the diagram in part D that we think is correct and match it with what we've found. So for section one, we know we start at 00. OK. We're gonna travel three hours. So in the X axis, we're gonna go over to the three and we're gonna travel 60 miles in that time. So in the long axis, we're going up to 60 we know it's gonna be a straight line because we have a constant velocity. And so that matches exactly with this first section of the graph in part D for section two. Now, we're starting where we left off, we're traveling for two hours. So we're starting at three hours, we're going up to five hours and we go a distance of 50 miles. We're starting from 60 miles. We add 50 miles, we end up at 100 and 10 miles just like the graph and part D. Ok. And again, a straight line because we have this constant speed. And then finally for this third segment, we are going four hours. So we're going from five hours up to nine hours. We're traveling 60 miles, we're starting at 100 and 10 miles. That means we go up to 170 miles and again, connecting with a straight line. And so that graph in option D does match exactly with what we've found based on the, the um slope. Ok. The sign of the slope based on the distance traveled and the speed and so the correct answer here is option D thanks everyone for watching. I hope this video helped see you in the next one.