The red and blue lines show the change in distance moved with time for two objects. Which color line corresponds to the object that moves the greater distance? (A) Blue, (B) red, (C) the objects move the same distance.
Okay, so in this question, we’re given a distance–time graph. That’s a graph that plots distance on the vertical or 𝑦-axis against time on the horizontal or 𝑥-axis. We can see that there are two lines drawn on this graph. There’s this blue line, which is a straight line that starts at the origin and ends at this point here. And then, there’s the red line, which also starts at the origin and ends at this same point. But we can see that this red line is not a straight line.
In this first part of the question, we are asked to work out which color line, blue or red, corresponds to the object that moves the greater distance. We know that both lines start at the origin, which is a distance value of zero. We also know that both lines end at the same point on the graph, which means that both lines end up at the same height on the distance axis. Now, the distance axis doesn’t have any units or a scale on it, so we can’t work out the actual distance traveled by either object. But we do know that both lines end up at the same height on this axis.
This means that whatever the particular value of distance that this height on the distance axis corresponds to, then at this point, both objects must have traveled that same amount of distance. We can see that this corresponds to the answer given in option (C). The objects move the same distance.
Now let’s move on and look at the second part of the question.
Which color line corresponds to the object that moves at the greater average speed? (A) Blue, (B) red, (C) the objects move at the same average speed.
Let’s begin by recalling that the average speed of an object is equal to the total distance moved by that object divided by the total time taken to move that distance. Now, we’ve already noted that on this distance–time graph, both the blue line and the red line start and end at the same points on the graph.
In the first part, we traced across horizontally from this end point of both lines until we got to the distance axis. Then, since both lines started at a height of zero on the distance axis and ended at this same height, we concluded that both objects had traveled the same total distance. So then, if we apply this equation for average speed to the objects represented by these lines on the graph, then we know that in the numerator on the right-hand side, this total distance will be the same for the object represented by the blue line and the object represented by the red line.
In the same way, we can trace vertically downward from this point on the graph until we get to the time axis. As with the distance axis, there’s no scale or units, so we can’t read off actual values of time. But we do know that since both lines start out at the origin with a time value of zero and both lines end up at this time value here that both objects travel for the same length of time.
This means that if we look again at this expression for the average speed of an object, then in the denominator of the right-hand side, this total time must also be the same for the object represented by the blue line and the one represented by the red line. If the total distance moved by each object is the same and the total time taken to move this distance is also the same, then the average speed in each case is equal to the same value in the numerator divided by the same value in the denominator. This means then that the average speed must be the same for both objects. This agrees with the answer given in option (C): the objects move at the same average speed.
Let’s now clear some space and look at the third and final part of this question.
Which color line corresponds to the object that has the greater maximum speed? (A) Blue, (B) red, (C) both objects have the same maximum speed.
In the second part of the question, we found that both objects had the same average speed. This meant that both objects took the same total amount of time from a time value of zero to this time value here to travel the same total amount of distance from a distance of zero to this distance value here. Now, in this third part of the question, we are no longer concerned with the average speeds, but rather with the maximum speed that each object reaches during its motion. Let’s recall that because a distance–time graph shows the distance moved by an object at each instant in time and because speed is defined as the rate of change of distance moved with time that the speed of an object at any instant in time is given by the slope of the corresponding line on a distance–time graph at that instant in time. The greater or steeper the slope, the greater the object’s speed.
In the distance–time graph from this question, we can see that the blue line is a straight line from the origin to this point up here. In other words, the slope of this line has a constant value the whole way along it. This means that the object whose motion is represented by this blue line must move at a constant speed. Now, let’s consider the red line on the graph. We know that this line has the same start and end points as the blue line and that this means both objects move with the same average speed. However, unlike the blue line, this red line is not a straight line, which means that it doesn’t represent a constant speed.
During the early part of the motion for small values of time, the slope of the red line is shallower or less steep than the constant slope of the blue line. So at these small time values, the object whose motion is represented by this red line is moving slower than the object whose motion is represented by the blue line, then towards the end of the motion. So for large values of time, we can see that the slope of the red line is now greater or steeper than the constant slope of the blue line. That means that by this point, the object whose motion is represented by the red line is now moving faster than the object whose motion is represented by the blue line. Overall then, this blue line represents some value of constant speed, while this red line represents a speed that starts out slower than this but ends up faster.
This means then that the line which corresponds to the object that has the greater maximum speed must be the red line. This is the answer given here in option (B).