The changes in the distance moved by an object during a time interval are shown in the graph. The graph is split into three sections, I, II, and III. In which section of the graph is the speed of the object greatest? A) I, B) II, C) III, or D) The speed is the same in all the sections.
So as you can see, we’ve been given a graph with time on the horizontal axis and the distance that an object has travelled on the vertical axis. And the blue line shows us how the distance that an object has moved changes as time passes. The question asks us to identify the section of the graph which shows the object travelling at the greatest speed. So a challenge in this question is to figure out how we can get the speed of an object if we’re just given a graph of the distances travelled against time.
There’s an important equation that will help us answer this question. 𝑠 equals 𝑑 over 𝑡; speed equals distance divided by time. We can think of this equation as being the definition of speed. If an object travels a certain distance 𝑑 in a certain time period 𝑡, then dividing the distance by the time it took to travel that distance will give us the speed 𝑠.
So let’s think about how we could apply this to section I on the graph. For the first part of the object’s journey, it starts at this distance and moves to this distance. In other words, it travels a distance equal to the difference between these two points. And the time it takes to travel this distance is given by the difference of these two points. So this vertical distance on our graph is the distance travelled by the object. And the horizontal distance shown on the graph is the time 𝑡 that the object took to travel that distance.
Unfortunately, the graph we’ve been given doesn’t actually have any measurements of distance or time on it, which means there’s no way of actually working out how far the object has travelled in section I, nor the amount of time that it took to travel this distance. In other words, because we can’t find values for 𝑑 or 𝑡, we can’t actually work out what the speed 𝑠 is. But even though we don’t have any actual values to plug into this equation, it still tells us something important about how the speed of an object affects its distance–time graph.
For example, let’s consider the distance–time graph for an object that travels a very large distance in a very short amount of time. That might look something like this. Now, in this case, we could work out the speed by taking the distance and dividing it by the time. The distance is very large and the time period is very small. So when we divide the distance by the time, we’d be dividing a large number by a small number, which will give us a large result. In other words, we’d say that this object is travelling at high speed. Let’s compare this to an object that travels a very short distance in a very long period of time.
The distance–time graph for such an object would look more like this. If we worked out the speed by dividing the distance by the time, then we’d be dividing a very small number by a big number, which would give us a small result. So we’d say that this object is travelling at low speed. What we find is that the faster an object travels, the steeper its distance–time graph becomes. This remains true even if the object is travelling in the opposite direction. So in these graphs, the distance is decreasing over time.
The steeper graph still shows a greater change in distance over a shorter period of time. So we know that the object shown in this graph is travelling at a higher speed than the object shown in this graph. We can summarize this in the following rule. The steeper the slope of a distance–time graph, the higher the speed. So, in order to establish which section of our graph shows the object travelling the fastest, we just need to determine which of these three line segments has the steepest slope.
We can see that section II of the graph has the gentlest slope. Section III is a bit steeper than that. And the graph is steepest in section I, which means the correct answer to this question is A. The speed of the object is greatest in section I.