The three graphs in the figures show the position of a particle, its velocity, and its acceleration against time, respectively. Identify each graph.
So we have three graphs here, one represented by the red plot, one by the blue plot, and one by the green plot. To figure out which graph is which, let’s begin by linking position or displacement, velocity, and acceleration. Let’s define the position of a particle at time 𝑡 to be equal to 𝑟, the velocity to be equal to 𝑣, and acceleration to be equal to 𝑎. We know that the velocity of an object is the rate of change of its position, so we can differentiate 𝑟 with respect to 𝑡 to find the velocity.
If we think about that graphically then, we know that the derivative of a function represents the slope of the function at that point. So we can say that the velocity must be the slope of 𝑟. Similarly, acceleration is rate of change of velocity with respect to time. And so we can say that the slope of our graph for 𝑣 must be the acceleration. So we’re going to need to begin by choosing one of our plots.
Let’s begin by looking at the red plot. It looks a little bit like a quadratic curve. It’s a parabola. We’re going to consider what happens to the slope of this curve. To begin with, the slope is negative. It remains negative for some time, but it gets less steep, which means the value of the slope must be increasing. At roughly three and a half time units, the slope is equal to zero. It then continues to increase. And so we can describe the slope of the red plot as being negative until 3.5 time units and then positive. And it appears to increase steadily.
So which of our plots represents this pattern? Well, if we look at the green plots, we see that it begins negative and it increases steadily until 3.5 time units, when it’s exactly equal to zero. And then it becomes positive. And we can say then that the slope of the red plot is the green plot. This still isn’t quite enough information, so let’s look next at the blue plot.
The slope of the blue plot is initially positive. It decreases until its slope is equal to zero at two time units. It then has a negative slope until it hits 𝑡 equals five, at which point its slope continues to increase and becomes positive. We therefore describe the slope of the blue plot as being positive until 𝑡 equals two, negative until 𝑡 equals five, then positive again. If we go to our graph, we see that this represents the red plot. We see that at 𝑡 equals two, the graph goes from being positive to negative and instantly passes through zero. Then at 𝑡 equals five, the reverse happens. And so we can say that the slope of the blue plot is represented by the red plot.
So how does this help us? Well, we know that the slope of 𝑟, the position, gives us 𝑣 velocity. Then in turn, the slope of 𝑣 gives us 𝑎, acceleration. The slope of the acceleration curve tells us nothing about the remaining curves. We saw that we didn’t need to consider the slope of the green graph, so the acceleration–time graph must be represented by the green plot.
We said that the green plot represents the slope of the red plot. So the velocity–time graph must be equal to the red plot. This leaves us with the distance–time graph as being represented by the blue plot.
And so we’ve identified each graph. The distance–time graph is blue, the velocity–time graph is red, and the acceleration–time graph is green.