In this video, we’re talking about graphing speed. This will involve looking at two-dimensional plots where the speed of an object is on the vertical axis and time is on the horizontal. Using a plot like this, we’re able to see what the speed of any of these three objects is at any moment in time.
To get started with this topic, let’s consider some of the basic shapes we might see on a speed-versus-time graph. On such a graph, we might see a horizontal line like this. Now, if we think about what this represents in terms of the motion of whatever object it corresponds to, we can see that over this interval of time, the speed of this object, whatever that speed is — we haven’t marked it out on our vertical axis — is staying the same. That is, the speed of the object at this moment in time is the same as it is at this moment and at this moment and at this moment and at any other moment on this interval. So a horizontal line segment on a speed-versus-time graph indicates an object moving with a constant speed. Now, that speed might be positive, like we’ve drawn it here, or it could be zero, like a line drawn here on the horizontal axis. But in either case, this basic graph shape of a horizontal segment indicates a speed that’s not changing.
Now, a horizontal line is one type of graph shape we could encounter on a speed-versus-time graph. Another shape is a line like this, which is sloping or pointing upwards. We can say that the gradient or the slope of this line segment is positive. And because of that, we can see that if we compare the speed at the start of this line segment with the speed at some later time, then we’ll find that the speed of our object has increased. And in fact, because the slope of this line segment is constant, that means that as we move left to right across it, the speed of our object will be increasing at a constant rate. So then, a line segment like this, with a positive constant slope to it, indicates a constantly increasing speed.
And then, a last type of graph shape we can consider is one that looks like this, a line segment with a constant negative slope. On a speed-versus-time graph, a segment like this indicates a speed that is decreasing, and decreasing at a constant rate. Now, there are other shapes that a line segment can take on a speed-versus-time graph. But these are three of the most common that we’ll encounter. One helpful way to think about these three line segments is in terms of their slope. Our first segment, because it’s a horizontal line, we know has a slope of zero. And that indicates a constant or unchanging speed. The second segment we know has a positive slope, and the third has a negative.
We could think of these three slope values as indicators of how speed changes over the line segment. A slope of zero indicates that the speed doesn’t change, a positive slope indicates that it increases, and a negative one corresponds to decreasing speed. Now, so far, we’ve been considering line segments on a speed-versus-time graph exclusively. But since the speed at which an object moves is related to the distance it travels, it can also be helpful to consider a distance-versus-time graph. Here, we’ve switched out the variable on our vertical axis, from speed 𝑠 to distance 𝑑. And once again, we have these three line segments, one with a slope of zero, another with a positive slope, and the last one with a negative slope.
Now, what, we wonder, can we say about these three segments on this new graph? Starting with the first segment, the horizontal line, we see that this corresponds to an object whose distance does not change over this time interval. If an object’s distance isn’t changing at all, that tells us something about the speed of the object. Object’s speed is equal to distance divided by time, so this object over this time interval must have a speed of zero. So when we see a horizontal line segment on a distance-versus-time graph, then that line segment corresponds to an object that is not in motion. And this is true, by the way, regardless of where that line segment appears. It could indicate different positive values of distance or even a value of zero. Regardless, so long as its slope is zero, then that means the speed of the object it corresponds to is also zero.
Now, what about a line segment that slopes upward on a distance-versus-time plot? We can see that for this line segment as time passes by, the distance that the object has traveled increases. We could imagine a scenario that corresponds to this. Say that this dot here is our object and that this initial moment in time for this line segment, the distance our object has traveled, begins to follow this orange line. So that would mean that our object down here has started to move in some direction. And the direction actually doesn’t make a difference.
But what is important about this object’s motion is that if we were to divide up this motion into even time intervals. Say that our object that starts here is here after the first time interval and then here after the second, here after the third, and so on. The distance that our object travels over each one of these time intervals must be the same. This is another way of saying that this object is moving at a constant speed. And moreover, that speed is not zero. It’s a positive value. And this tells us how a line segment with a constant positive slope on a distance-versus-time graph tells us about corresponding object’s speed. An object with a line segment like this is moving at a constant positive speed.
Now, let’s move on to consider this last line segment, the one with the negative slope. When this segment was plotted on a speed-versus-time graph, we saw that it indicated a constantly decreasing speed. But now that it’s on a distance-versus-time graph, let’s consider what it might mean. Down here, we have our object in motion. And we know that the total distance this object travels is equal to the sum of the total ground that it covers. That means if our object followed a path like this, going out to the right then down, then back to the left, we would find its total distance traveled by walking all that distance out, so to speak, the whole length of the path it travels.
So rather than this section of the path, say, being positive distance and this section of the path corresponding to negative distance, rather, any distance traveled increases the total distance that our object has moved. And we can’t undo distance that we’ve traveled. Say our object was at this point and then we doubled back and covered this ground again. That wouldn’t remove this segment from our total distance traveled, but would rather increase that total distance because we’ve now moved along that path twice.
All this to say, when an object is in motion, there’s no such thing as decreasing its total distance. Motion always increases distance traveled, and that brings us to this line segment. We can see that at the initial time value of this line segment, the total distance indicated is greater than the distance indicated at a later moment in time on this segment. This is saying then that somehow the distance this object has traveled has decreased. But as we’ve seen, that’s not how distance works. So long as an object is moving, its distance is always increasing.
So then, even though a line segment like this is possible and makes perfect sense on a speed-versus-time graph, we’ve seen that it’s not physically possible on a distance-versus-time graph. That comes down to the definition of what distance is. Distance is the sum of all movements of an object. Therefore, distance can never decrease. And therefore, a line like this, with a negative slope, can never appear on a distance-versus-time graph.
Now that we’ve seen speed-versus-time and distance-versus-time graphs separately, let’s see how we can create them both for the same given scenario. Say that we have an object here that’s moving along at a constant speed. It keeps this up for some amount of time. And then, the object’s speed begins to increase, and it does so at a steady rate. After some other amount of time though, the object’s speed begins to decrease. And again, it does this at a steady rate until its final speed is equal to the speed it started out with. We can plot this object’s motion on a speed-versus-time and a distance-versus-time graph.
To start doing this, let’s consider that the object’s motion is basically divided into three segments. The first segment is the one where it has a constant speed. The second segment of time is the one where its speed is increasing at a steady rate. And in the third segment is the one where its speed is decreasing, again at a constant rate. So for our first interval of time, and we’ll say that this interval begins at time 𝑡 equals zero, our object is moving at a constant speed. We don’t know exactly what the speed is, but we know that it’s not zero, that the object truly is moving. This means we can go to our speed-versus-time graph. And we’ll plot a horizontal line segment corresponding to this object’s speed for this first time interval.
Now, how do we translate this speed-versus-time line segment to a corresponding line segment on a distance-versus-time graph? Well, we can recall that when an object moves with a constant nonzero speed, that means that its distance is increasing at a constant rate. And let’s say that the distance our object has traveled before it starts moving is zero. In other words, on our distance-versus-time graph, we’ll begin at the origin, where distance and time are both zero. Now, since our object is moving along with a constant speed, and we’ve seen that that indicates a distance that’s increasing at a constant rate, then the distance-versus-time graph for this segment will look something like this. A positively sloping line with a constant slope and lasting the same amount of time as our speed-versus-time segment.
Okay, so we’ve covered the first time interval of our object’s motion. Now, let’s consider the second interval where it’s moving with a steadily increasing speed. Now, for a speed that’s increasing at a steady rate, we know that means that on our speed-versus-time graph, this line segment will have a positive slope to it, and the slope will be constant. Depending on just how fast the object’s speed is increasing, that segment could look like this. This part of our graph with a constant positive slope corresponds to a constantly increasing object speed. And now, we can figure out how to plot on our distance-versus-time graph a line segment that corresponds to this one. That is, when an object’s speed is constantly increasing, what will that object’s distance traveled versus time look like?
Here’s one way we can think of this. When we had an object whose speed was constant and positive, we saw that the distance that object traveled increased at a steady rate. Now, we have an object whose speed is increasing at a steady rate. And what that will lead to is a distance-versus-time curve that increases not at a steady rate, but at an increasing rate. So if we consider the point in time that begins this second line segment, the one where our speed is constantly increasing, then we know that, at that instant, the total distance our object has traveled is indicated by this point here on our distance-versus-time line segment.
Now, if our object’s speed, instead of starting to increase, were to maintain this constant value that it started out with, then we know what our corresponding distance-versus-time curve would look like. It also would continue on in line with our original line segment. But of course, the speed of our object doesn’t stay constant. But rather, it begins to increase. This tells us that instead of covering distance at this steadily increasing rate, as our speed increases along this line segment, we’ll be covering distance at a faster rate. In other words, our distance-versus-time segment, corresponding to this constantly increasing speed, will look something like this. It will be a line that curves upward.
Now, it turns out that there’s a mathematical connection between a distance-versus-time graph and a corresponding speed-versus-time graph. To see that connection, let’s consider these four different points along the curved portion of our distance-versus-time graph. The slope of this curved line at each one of these points is different. As we move from left to right along this curved portion, the slope of the line as we move in this direction increases. And notice that as we move over the same time interval on our speed-versus-time graph, the object’s speed increases as well. And this is where we see the connection between distance and speed. The rate at which the distance of an object changes, where that rate is indicated by the slope of the distance-versus-time graph, is actually equal to the speed of that object at that instant in time.
So this point here, with this particular slope, corresponds to a speed value here, whereas this second point here, with a greater slope to it, corresponds to a greater speed. And this trend continues as we consider our third point with an increased slope and corresponding increased speed. And then also with our last point, which has the greatest slope, and this corresponds to the greatest object speed. It’s this connection between distance and speed that lets us know that this segment of our distance-versus-time curve is correct. It corresponds to a steadily increasing object speed. So that’s our second time interval. And now, let’s consider our third and last one, where our object’s speed is steadily decreasing.
When we consider what this looks like on our speed-versus-time graph, we can recall that a line segment on such a graph with a constant negative slope indicates a constantly decreasing object speed. And since in this case we know that our object ends up with the speed that it started with in the first place, this line segment would look something like this. So here, at the end of our third time interval, our object attains the speed it started out with. Now, when we consider what this segment will look like on our distance-versus-time graph, we might think that since speed is decreasing with time, distance will as well. But it’s important to remember that distance never can decrease, as we saw earlier.
Even though the speed of our object is decreasing over this last time interval, since the object’s speed is always positive throughout that interval, that means the distance that our object travels will continue to increase. Here’s another way to think of it. Let’s say that for this third time interval, instead of our speed-versus-time graph looking like this, say that our object dropped suddenly to a speed of zero and maintained that speed for the rest of the interval. Zero is the least amount of speed our object can possibly have. But even if that were to happen, our corresponding distance-versus-time line segment would not have a negative slope. Instead, it would look like this. The distance would simply stay constant over this interval.
Now, since our object’s speed is not zero but rather behaves like this over this time interval, we know that the corresponding distance-versus-time line segment will not have a slope of zero, but will instead have a positive slope. To see what this segment will look like, let’s recall that the value of our object’s speed at various moments in time is equal to the slope of our distance-versus-time curve at those same instants. So all our speed value is up here that, say, corresponds to a slope like this on our distance-versus-time curve, whereas this relatively lower speed value corresponds to a relatively flatter slope. And this smaller speed value yet corresponds to a yet flatter but still positive slope. And even more so for this final speed value of our object, this corresponds to a slope that is still positive because the speed is not zero. But it’s closer to our horizontal line than any of these previous tangents.
So now we can see the general shape that our distance-versus-time curve for this interval of time will take. It will look a bit like this. Note that, across the total time interval of this object’s motion, while its speed increased and decreased throughout this time, the total distance traveled by the object only increased. And we’ll find that to be true in general. No matter how object speed behaves, object distance traveled can only increase or stay constant; it can’t decrease.
Let’s summarize now what we’ve learned about graphing speed. Starting off, we looked at three common line segments that appear on speed-versus-time graphs. We saw that a horizontal line segment corresponds to an object’s speed that is constant, while a line segment with a constant positive slope corresponds to an object whose speed is steadily increasing. While a line segment with a constant negative slope indicates an object whose speed is constantly decreasing.
Next, we considered how similar line segments on a distance-versus-time graph might tell us about object’s speed. We saw that a horizontal line segment on a distance-versus-time graph indicates an object whose speed is zero, while a line segment with a constant positive slope indicates an object with a constant positive speed. And then, we saw that a line segment with a constant negative slope on a distance-versus-time curve is not possible. This is because such a line segment would indicate a decreasing distance, while distance always increases or remains the same.
Lastly, as we studied object motion and the corresponding speed and distance-versus-time graphs, we saw that the slope of a distance-versus-time graph at a particular moment in time is equal to the object’s speed at that same instant in time. This is a summary of graphing speed.