2 – Nd113 C6 L1 09 L Interpreting Position Vs Time Graphs V3

Data often tells a story. And often, a good plot can really help reveal what that story is. Now I’d like to show you how I think about these position versus time graphs to figure out the story the data tells. Here, we have two axes. In this case, time is the horizontal X axis and position is the vertical Y axis. And to begin, let’s start with the simplest version of our story by adding two data points, one for a starting position of 30 miles at 2:00, and one for ending position of 80 miles at 3:00. Now, whenever I see data like this, it’s often helpful to imagine or even actually plot a line that connects the points, and this line represents where the car would have been at every moment between 2:00 and 3:00 if it were traveling at a single constant speed the whole time. The line like this can be pretty helpful in making sense of these plots. Because once I have the line, it’s easier to see the change in position, or delta X, which is just the vertical separation between the two points. And change in time, or delta T, which is just the horizontal separation of these points. And as you saw earlier, you can also calculate average speed by dividing delta X by delta T. Now, I want to point something out that has some pretty profound implications for calculus. The calculation of V average is the same calculation you would make to find the slope of this line. If you remember back to algebra, slope is calculated as the vertical change in the graph divided by the horizontal change. Or as I learned it, slope equals rise over run, which in this case, is delta X divided by delta T, so you can see that the slope of this line gives us the average speed between these two points. And when the slope is steeper, it means the average speed is faster. Now, we have to be careful to remember that this line is really just something we drew. It does not necessarily reflect where the car really was during the whole trip. If we want to know a more complete story, we’re going to need more data. So, let’s say we had sampled at 20-minute intervals and found that at 2:20, the odometer read 40, and at 2: 40, it read 70. Well, once again, I’m going to draw some lines to connect these dots. And now, remembering that the slope of the line between any two points gives the average speed between those points, I immediately can see that here in the first 20 minutes, the car was on average going much slower than it was in the next 20 minutes since the slope isn’t nearly as steep. Then, in the last 20 minutes, the car, on average, was moving slow again, and this really helps make the story clearer. And if I want an even better understanding, I would need even more data sample with an even shorter time interval, like every 10 minutes or 5 minutes. And with this much data, we don’t even really need the connecting lines as a visual cue. And now, I can see that the average speed appears to have been greatest between 2:30 and 2:35, and they know that because I can see that the slope is steepest here, which just means that delta X over delta T is greatest here. And now, I like to give you a chance to interpret some graphs on your own.