In this video, we’re looking at kinematic graphs. Kinematics is a branch of mechanics that looks at the motion of objects without thinking about the forces that act on them. So common kinematic quantities include things like acceleration, velocity, and displacement. When we talk about kinematic graphs, we generally mean graphs that show how kinematic quantities, such as these, vary over time. So in this video, we’ll be looking at how we can interpret graphs of various kinematic quantities against time. And specifically, we’ll be looking at how we can get extra information by looking at the gradient of these graphs and the area under them.
So first, let’s consider a graph of displacement against time. Here, we have the displacement 𝑠 on the 𝑦-axis and the time 𝑡 on the 𝑥-axis. This graph shows us an object whose displacement is initially zero. We can see that as time passes, its displacement increases at a constant rate before reaching some maximum displacement and staying there for some time. The displacement then decreases rapidly, becoming negative at this point before reaching a maximum displacement in the negative direction. Finally, the object returns to the zero displacement position, but this time at a varying rate.
Now, as well as providing us with information about the displacement of the object at any given time, this graph also enables us to find out other kinematic quantities. Specifically, the gradient or slope of the graph at any point tells us the velocity of the object at that time. For example, between the time zero and this time, we can see that the graph has a constant positive gradient. From this, we can determine that between these times, the object was traveling in the positive direction with a constant velocity. And calculating the exact value of this gradient will tell us exactly what the velocity is.
Similarly, between these two times, we can see that there’s a constant negative gradient. This tells us that for this period, the object has some constant negative velocity. Even if we don’t calculate exactly what this gradient is, we can see by inspection that this line segment is steeper than this line segment, which tells us that the speed of the object was greater here than it was here.
Finally, if we look at the graph between this time and this time, we can see that it’s curved. In other words, the gradient is changing. Initially, the graph is sloped gently in the positive direction, and as time goes on, we can see that the graph gets steeper. The curved graph means it’s more difficult to exactly calculate the gradient and therefore the velocity at any point in time. But by considering the overall changes in the gradient, we can see in this case that the object goes from a small positive velocity to a large positive velocity. And calculating the gradients of these tangents to the graph will tell us the value of the velocity at those times.
Now, generally speaking, the gradient or slope of a graph is given by dividing a change in the 𝑦-variable by a change in the 𝑥-variable. And we can think of this as describing the rate at which the 𝑦-variable changes when we change the 𝑥-variable. In this case, the 𝑦-variable is displacement 𝑠 and the 𝑥-variable is time 𝑡. So the gradient of the graph at any given point is equivalent to a change in displacement divided by a change in time. We might recognize this as a formula for velocity. In other words, the gradient of a displacement–time graph gives us the rate of change of displacement with time.
At this point, we can note the similarity between a displacement-versus-time graph and a distance-versus-time graph. For any given motion, these two graphs sometimes look the same, but it’s important to remember that while displacement is a vector quantity, distance is a scalar quantity. This means that distance is always positive, but displacement can be positive or negative depending on the direction.
This is what the distance–time graph looks like for the same motion. We can see that the first part of both our distance–time and our displacement–time graphs look the same. But negative values of displacement are represented as positive values of distance on our distance–time graph. Finding the gradient at any point on a distance–time graph also tells us how fast an object was traveling at that time. But because distance is scalar and doesn’t include any information about direction, finding the gradient or slope will actually just tell us the speed, the scalar quantity, rather than the velocity, which is a vector quantity.
In practice, we just need to remember that the sign of the gradient, that is, whether it’s positive or negative, is important on a displacement–time graph. A positive gradient means a positive velocity and a negative gradient means a negative velocity. However, on a distance–time graph, we’re only interested in the magnitude of the slope. This tells us the magnitude of the velocity and therefore the speed. For our next kinematic graph, let’s take a look at a velocity–time graph.
Here, once again, we have time on the 𝑥-axis, but we have velocity 𝑣 on the 𝑦-axis. We can see that this graph describes the motion of an object whose velocity increases at an increasing rate before decreasing at a constant rate. The velocity then becomes negative and reaches a maximum value in the negative direction, which it maintains for some time before returning at a constant rate to zero velocity.
Once again, it’s possible to obtain information about another kinematic quantity by looking at the gradient of this graph. The gradient or slope is equivalent to a change in the 𝑦-direction divided by a change in the 𝑥-direction, which in this case would be a change in velocity divided by a change in time. And we can recognize this as the formula for acceleration, which tells us that the gradient or slope of a velocity–time graph gives us the acceleration. Both velocity and acceleration are vector quantities. So once again, it’s important that we consider whether the gradient is negative or positive. A negative gradient corresponds to a negative acceleration, that is, an acceleration in the negative direction, whereas a positive gradient corresponds to an acceleration in the positive direction.
Both of these parts of the graph are straight lines. In other words, they have a constant gradient, which means that the acceleration takes a constant value. However, this part of the graph is curved. A curve means the gradient is changing. Therefore, in this case, the acceleration is changing. And by drawing tangents to the graph at different times, we can see that the gradient is increasing and therefore the acceleration is increasing over this period. As well as telling us about acceleration, velocity–time graphs can enable us to calculate another kinematic quantity. Specifically, the area between the graph and the time axis tells us the displacement. So, for example, between the time zero and this time, the change in displacement of the object is given by this area.
It’s important to note that areas, such as this, above the time axis correspond to a positive change in displacement, whereas areas below the time axis, such as this, correspond to negative changes in displacement. So throughout this time interval, the object was being displaced in the positive direction. And throughout this time interval, the object was being displaced in the negative direction. The total change in displacement of the object can be obtained by adding together this positive displacement and this negative displacement.
In this case, we can see that the area corresponding to negative displacement is larger than the area corresponding to positive displacement. This means that when we add together the total positive displacement and the total negative displacement, we’ll end up with a negative answer. In other words, the overall displacement of the object throughout the entire journey is negative. We can also say that at the end of the journey, the object will have some negative displacement.
Velocity is another vector kinematic quantity that has a corresponding scalar quantity, in this case, speed. Once again, parts of these graphs are identical to each other. But we can also see that negative values of velocity on the velocity–time graph are represented by positive values of speed on the speed–time graph. Since speed and velocity are conceptually similar but speed doesn’t contain any information about direction, we can use the same kinds of rules for gradient and area of a speed–time graph. But we wouldn’t be able to determine any information about the direction of these quantities. In other words, we could say that the magnitude of the gradient or slope on a speed–time graph tells us the magnitude of the acceleration. And the magnitude of the area under a speed–time graph tells us the magnitude of the displacement.
We can recall that the magnitude of displacement is equal to the scalar version of displacement, in other words, distance. For some reason, we don’t really have a special name for the scalar version of acceleration. In fact, sometimes this just gets called acceleration. So it’s important to remember that when we’re dealing with a speed–time graph, the gradient can’t tell us the direction of the acceleration. Now, to help us remember what quantity is represented by an area on a graph, it can be helpful to consider a square or a rectangle on that graph. For example, let’s consider the area of this rectangle on the velocity–time graph. We know that the area of a rectangle is given by its height multiplied by its width. But what quantity does this represent?
Well, on this graph, the height extends in the vertical direction, which means it represents a velocity. Similarly, the width extends in a horizontal direction, and so it represents a time. This means that the area, which is a height multiplied by a width, must represent a velocity multiplied by a time, and velocity multiplied by time is equivalent to displacement. And this shows us that any area on a velocity–time graph corresponds to a displacement. And similarly, any area on a speed–time graph corresponds to a distance.
For a final example, let’s now look at an acceleration–time graph. In this case, we have time on the 𝑥-axis once again and we have acceleration 𝑎 on the 𝑦-axis. This graph shows us the motion of an object which initially has some positive acceleration, which it maintains for some time before decreasing at an increasing rate until it reaches zero acceleration. The object then has zero acceleration for some time before its acceleration starts to increase in the negative direction at a constant rate. It then reaches a maximum negative acceleration, which it maintains for some time, and the acceleration then moves back to zero at a constant rate. Once again, we can calculate other kinematic quantities by looking at the gradient of this graph and the area under the graph.
Once again, the gradient or slope is equivalent to a change in the 𝑦-variable divided by a change in the 𝑥-variable. In this case, that’s a change in acceleration divided by a change in time. We could describe this as the rate of change of acceleration. And while this is sometimes a useful quantity to think about, it’s not often that we actually need to calculate this. So we won’t really be concentrating on this quantity in this video. But what we will be thinking about is the area under this graph. The area under an acceleration–time graph corresponds to a change in velocity. Now, acceleration is a vector quantity, so direction is important here. An area above the axis can be thought of as positive. It represents a positive change in velocity, whereas an area below the axis corresponds to a negative change in velocity.
In other words, throughout this period on the graph, the velocity is changing in the positive direction. And throughout this period on the graph, the velocity is changing in the negative direction. We can see in this case that the positive area is larger than the negative area. This means that the overall change in velocity in the positive direction is greater than the overall change in the negative direction, which means that across the entire duration of the journey, the velocity will overall change in the positive direction.
To help us remember what area represents on an acceleration–time graph, we can once again consider the area of a rectangle on this graph. We know that the area of a rectangle is given by its height multiplied by its width. And on this graph, the height, which is measured in the vertical direction, corresponds to an acceleration, while the width is measured in the horizontal direction, so it must correspond to a time.
This means that the area of this rectangle represents an acceleration multiplied by a time. And an acceleration times a time gives us a velocity. This shows us that any area on an acceleration–time graph corresponds to a velocity. We’ve now seen how looking at the slopes and areas on a range of kinematic graphs can give us different kinematic quantities. On a displacement–time graph, we’ve seen that the gradient tells us the velocity. And on a velocity–time graph, the gradient tells us the acceleration. We’ve also seen that areas on velocity–time graphs correspond to changes in displacement, while an area under an acceleration–time graph corresponds to a change in velocity.
Now, displacement, velocity, and acceleration are all vector quantities. But similar rules apply for graphs of the scalar versions of these quantities. For example, the scalar version of a displacement–time graph is a distance–time graph. And the gradient of this tells us the speed, which is the scalar version of velocity. Similarly, we could say that the scalar version of a velocity–time graph is a speed–time graph. The gradient of such a graph would tell us the magnitude of acceleration, which is more or less the scalar version of acceleration. And the area under a speed–time graph would tell us the change in distance, which is the scalar version of displacement.
Let’s now put this information into practice by looking at a question.
Which of the following quantities is equal to the gradient of the type of graph shown? (A) Speed, (B) acceleration, (C) distance, (D) velocity, or (E) change in velocity.
Looking at this graph, we can see that the vertical axis is labeled displacement and the horizontal axis is labeled time. And we’re being asked to find out what quantity is equal to the gradient or slope on this graph. Now, this graph is incomplete. There’s no line on it, so we can’t actually calculate a gradient. However, we just want to determine what the gradient represents on this type of graph. So it’s not actually important what the graph looks like.
To help us figure out the answer to this question, let’s think about how we would calculate the gradient. Generally speaking, the gradient or slope of a graph is equal to a change in 𝑦 divided by a change in 𝑥. So if our graph was a straight line, we could pick any two points on that line and calculate the gradient by dividing the vertical difference between these points by the horizontal difference.
Now, this is true for all graphs. However, on this specific set of axes, any vertical distance on the graph actually corresponds to a displacement, while any horizontal distance on the graph corresponds to a time. This means that when we’re calculating a gradient on this graph, we’re effectively dividing a change in displacement, 𝛥𝑠, by a change in time, 𝛥𝑡. So we’ve shown that the gradient is equal to this quantity. Now we just need to find which of our answer options this quantity corresponds to.
To help us do this, we can look at the dimensions of this quantity. The dimensions of a change in displacement over a change in time, which we can represent using square brackets, are equal to the dimensions of a change in displacement divided by the dimensions of a change in time. Since displacement has dimensions of length, this means that a change in displacement also has dimensions of length. And similarly, the dimensions of a change in time is just time.
So we’ve shown that the gradient on this graph has dimensions of length divided by time. Alternatively, it can be helpful to think about the units that we might use to represent the gradient. The standard unit for representing displacement is the meter, and the standard unit for representing time is the second. Since the gradient is equivalent to a change in displacement divided by a change in time, this means that the gradient could be represented in units of meters per second. So we’ve now determined that the gradient is a quantity which has dimensions of length over time. Or, in other words, it’s a quantity that can be measured in meters per second.
Looking at our answer options, we know that the same can be said of speed, velocity, and change in velocity. But the same cannot be said of acceleration and distance, which means that (B) and (C) are not correct answers. Furthermore, we can recall the displacement is a vector quantity, meaning it has both magnitude and direction. Since the gradient is equal to a change in this vector quantity divided by a change in a scalar quantity, we can conclude that the quantity represented by the gradient must also be a vector quantity. Velocity and change in velocity are both vector quantities. However, speed is not; it’s just a scalar quantity, so we know that this is not the correct answer either.
So we now have two answer options left to choose from. To help us choose between them, we can note that the gradient of a line has a specific value at any given point. If our displacement–time graph is a straight line, the gradient is the same at every point on the line. And if our displacement–time graph is curved, then it takes different values at different points. But regardless of the shape of our displacement–time graph, the gradient has a specific numerical value at any given instant in time.
This tells us that the gradient can never give us the change in velocity because in order to describe a change, we would need to compare the velocity at two different times. Instead, the gradient at any given time on a distance–time graph is equal to the velocity of the object shown at that time. This means the correct answer is (D). The quantity equal to the gradient on a displacement–time graph is velocity.
Let’s now summarize the key points that we’ve looked at in this video. In this video, we’ve seen that when we’re looking at graphs of various kinematic quantities against time, we can calculate other kinematic quantities by looking at the gradient or slope of the graph and the area between the graph and the time axis. Specifically, the gradient of a displacement–time graph gives us the velocity of an object, and the gradient of a velocity–time graph gives us the acceleration of an object. The area on a velocity–time graph can tell us the change in displacement of an object, and the area on an acceleration–time graph can give us the change in velocity of an object. And these same relationships exist with corresponding scalar quantities as well. This is a summary of kinematic graphs.