Question Video: Using Distance-Time Graphs to Compare the Speeds of Two Objects | Nagwa Question Video: Using Distance-Time Graphs to Compare the Speeds of Two Objects | Nagwa

# Question Video: Using Distance-Time Graphs to Compare the Speeds of Two Objects Science • Third Year of Preparatory School

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Two distance–time graphs show objects moving with uniform speeds. How do the speeds of the objects compare?

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### Video Transcript

Two distance–time graphs show objects moving with uniform speeds. How do the speeds of the objects compare?

In this question, we’ve been given two distance–time graphs for two different objects, and we want to compare the speeds of the objects. Is the speed of the object shown by the blue or the red line greater? Or are they equal? To answer this, let’s first take a look at our graphs and pick out some important information. Because they’re both distance–time graphs, each of them has a vertical axis showing the distance in meters that the object has traveled and a horizontal axis showing the time in seconds that the object has traveled for. Overall, these graphs look very similar, but notice that they actually have different scales along their axes.

Look at the graph on the left. Here on either axis, the side length of each grid square represents 10 units. So the distance axis reads zero meters, 10 meters, 20 meters, and so on, and the time axis reads zero seconds, 10 seconds, 20 seconds, and so on. But the graph on the right uses different scales. On this graph, the side length of each grid square represents one unit on either axis. So this distance axis reads zero meters, one meter, two meters, and so on, and the time axis reads zero seconds, one second, two seconds, and so on.

It will be important to keep these differences in mind when we compare the two graphs. Now to answer this question, we need to compare the speeds of the two objects represented by the lines on these graphs. Recall that on a distance–time graph, the speed of an object is equal to the gradient of the line that represents its motion. Therefore, to compare the speeds of the objects, we’ll calculate and compare the gradients of the two lines.

Now, we might recall that when multiple lines are drawn on graphs that use axes with the same scales, we can compare the gradients of the lines just by looking at them. For example, if we wanted to compare the gradient of this blue line to, say, this pink line, we would know that the pink line represents a faster-moving object since the pink line is steeper than the blue one. But we would only know that for sure because the pink and blue lines are plotted using the same scales on the distance and time axes.

For lines on graphs that have different scales though, like the two graphs in this question, we can’t necessarily compare their gradients visually. It’s much safer to calculate the gradient for each line and then compare those values. To do this, we should recall that we measure the gradient of a line between two points on the line and that the gradient is equal to the change in distance divided by the change in time between those points.

We already know that the gradient of the line corresponds to the speed of the object. So we’ll use this formula as our formula for speed. Also note that because both of these objects are moving at uniform or constant speeds, both of the lines on the graphs have constant gradients. This means that for each graph, we can choose to measure the gradient between any two points along the line of motion.

Now that we understand how to use this formula, let’s apply it to each graph, and we’ll use the subscripts blue and red to express which line we’re dealing with. Let’s start with the graph on the left with the blue line of motion and measure the object’s speed or gradient between this point and this point.

First, let’s find the change in distance between the points. We can see that this point corresponds to a distance of zero meters, and this point corresponds to a distance of 40 meters. Therefore, the change in distance between these two points is equal to 40 meters minus zero meters, which is just 40 meters. Next, we’ll find the change in time between the two points. We can see that this point here corresponds to a time of zero seconds, and this point corresponds to a time of 40 seconds. So the change in time between these two points equals 40 seconds minus zero seconds or just 40 seconds.

Now, we can substitute these values into the equation for the speed of the object. So for the blue line, we have a speed of 40 meters divided by 40 seconds, which simplifies to one meter per second. This is the speed of the object represented by the blue line. Now, we just need to repeat this process for the graph on the right with the red line of motion.

Let’s choose these two points to measure the speed between. And we’ll start by working out their change in distance. We can see that this point corresponds to a distance of zero meters, and this point corresponds to a distance of four meters. So the change in distance between the points equals four meters minus zero meters, which is just four meters.

Next, we’ll find the change in time between these two points. This point here corresponds to a time of zero seconds, and this point here corresponds to a time of four seconds. Thus, the change in time between these two points equals four seconds minus zero seconds, or just four seconds. Now, let’s substitute these values into the equation for the speed of the object represented by the red line, and we get a speed of four meters divided by four seconds, which simplifies to one meter per second.

So we’ve calculated that both of the objects move at a speed of one meter per second. And therefore, when asked how the speeds of the objects represented by the lines on the graphs compare, we know that the speeds of the objects are equal.

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