The thinking distance and braking distance for a car at different initial speeds are shown by the lengths of the two-coloured bars in the diagram. The longer the bar, the greater the initial speed the car stops from. Which of the following quantities is shown by the length of the grey part of the bar? a) Stopping distance, b) thinking distance, c) braking distance.
Alright to answer this question, we need to work out what the grey parts of these bars represent — whether that’s stopping distance, thinking distance, or braking distance. And the other important thing that we know is that the longer the bar, the greater the initial speed the car stops from. The initial speed gets larger as we go down the diagram. So this first one is a low initial speed, this is slightly higher, and so on until we get to the highest one that’s on the diagram.
Now that’s all of the important information from the first part of the question. So let’s erase it and have a think about options a to c and what they mean. Firstly, let’s think about stopping distance.
Stopping distance is the total distance travelled by a car while it comes to rest. In other words, it’s the thinking distance plus the braking distance. That’s the total distance travelled by the car when the driver realizes they have to brake plus the distance travelled by the car when the brakes have been applied and the car is slowing down. Now, stopping distance consists of two distances: the thinking distance and the braking distance. So it cannot be represented by only the grey part of the bar. In fact, the stopping distance in this case is represented by the entirety of the length of the bars, whereas we only want what the grey part represents. Therefore, stopping distance is not the answer to our question.
Now, options b and c are thinking distance and braking distance, respectively. So the grey part of the bar could be representing either one of the things that make up stopping distance. But which one is it? The thinking distance of a car is the distance travelled by the car, while the driver reacts to some external stimulus. For example, if a pedestrian walks out into the road in front of the car, the driver needs some time to react to this. That’s the driver’s reaction time. And in this period of time, the car is still moving. So it travels some distance. That distance is known as thinking distance.
Now, the important thing to know is that the braking distance scales linearly with initial speed. In other words, the thinking distance is directly proportional to the initial speed of the car. Moving onto braking distance, this is the distance travelled by the car when the brakes have been applied and the car is slowing down coming to a stop.
In other words, the driver has seen the pedestrian walk out into the road in front of them, they’ve reacted and travelled the thinking distance, and then they’ve stepped on the brakes. At this point, the car stops slowing down, but it’s still moving. So it moves a certain distance. This distance is the braking distance and that is measured once the brakes have been applied until the car comes to rest. Now, the important thing to know about braking distance is that it scales quadratically with the initial speed.
So there’s an important relationship that we need to remember here. Thinking distance scales linearly with the initial speed and braking distance scales quadratically with the initial speed. We can use this to work out which of these two distances is represented by the grey part of the two-coloured bars. A simpler way to write this relationship is that the thinking distance is directly proportional to the initial velocity, which we’ll call 𝑣, whereas the braking distance is proportional to 𝑣 squared. That’s the quadratic relationship.
So with that in mind, we can make two different plots showing the lengths of the purple bars and the grey bars as the initial velocity increases. On the horizontal axes, we’ll have the initial velocities 𝑣. And on the vertical axes, we’ll have the distances represented by the lengths of the purple and grey bars. Now since we’re trying to find out what the grey bars represent, let’s plot those first.
Starting with the lowest initial velocity, so we’re representing this grey bar at a low initial velocity, which let’s say looks something like this. The next bar at a slightly higher initial velocity looks something like this. Plotting the third one looks something like that. And plotting the final two makes the graph look like this.
Now since we know that in this diagram the grey bars are starting at the same point, we could’ve equally well just compared the lengths of these bars on this diagram itself rather than plotting a graph. However, the advantage of drawing a graph is that we can compare the lengths of these bars and we can draw a line through them. Now, this line looks like a straight line.
Of course, my drawing here is not accurate. This line would normally go through the origin, but we can do the same thing on the diagram itself. We can draw a straight line through the bars. Applying the same logic to the purple bars on the second graph, we can see that when we compare the lengths of these bars and draw a line through them, we get what looks like a quadratic curve. And once again, we can do the same thing on the diagram.
Now, the reason that we plotted these two curves up here — this one and this one — is because then we can clearly see what the axes represent. We can see that the horizontal axis represents the initial velocity and the vertical one represents the distance travelled, whereas in the diagram down here, we just had to remember that the length of the bar represents the distances and that the initial velocity was increasing as we went down the diagram.
However, the principle is the same. We realize that the purple bars — these parts of the bars — represent a quadratic relationship as the initial velocity increases, whereas the grey parts of the bar represent a linear relationship. And hence, the grey parts of the bar do represent the thinking distance because a thinking distance is directly proportional to the initial velocity, whereas the purple parts of the bars represent the braking distance because that is proportional to the square of the initial velocity.
In other words, what we discovered is that the grey bars increase in a linear fashion. Therefore, they must be the thinking distance. And hence, our final answer is that the grey parts of the bars represent the thinking distance.