Video Transcript
Do the speeds corresponding to the lines shown on the following distance–time graph change value equally for any two adjacent lines?
Alright, so in this question, we’re being 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 four different lines drawn on this graph. And in this question, we’re being asked to compare the speeds corresponding to each of these lines. Let’s label the speed corresponding to the blue line as 𝑉 subscript b, the speed corresponding to the red line as 𝑉 subscript r, the green line 𝑉 subscript g, and the orange line 𝑉 subscript o. We’re being asked if the speeds change value equally for any two adjacent lines on the graph. If the speeds do change value equally for any two adjacent lines, then that means that the difference between the speeds of each pair of adjacent lines must be the same.
From the graph, we can see that the blue line and the red line are adjacent. And the difference between their speeds is 𝑉 subscript b, the speed corresponding to the blue line, minus 𝑉 subscript r, the speed corresponding to the red line. Now, the red line is also adjacent to the green line. So, if the speeds change value equally for any two adjacent lines, then the difference between the blue and the red line speed, that’s 𝑉 subscript b minus 𝑉 subscript r, must be equal to the difference between the red and the green line speeds. That’s 𝑉 subscript r minus 𝑉 subscript g.
Then, in the same way again, we can see that the green line and the orange line are also adjacent. So if it’s true that the speeds change equally between adjacent lines, then 𝑉 subscript g minus 𝑉 subscript o, that’s the difference between the green and the orange line speeds, must also be equal to these other two speed differences.
To answer this question then, we need to work out these three differences between the speeds. Then, we can compare these three differences to see whether or not this statement is true. If we find that it is true, then we know that the speeds do change equally between any two adjacent lines. Conversely, if we find that the statement isn’t true, then we know that the speeds don’t change value equally between adjacent lines.
In order to work out these three differences, let’s begin by finding the values of the four different individual speeds. Let’s recall that the speed of an object is defined as the rate of change of the distance moved by that object with time. This means that if an object moves a distance of Δ𝑑 and it takes a time of Δ𝑡 in order to do this, then the average speed of that object, which we’ll label as 𝑉, is equal to Δ𝑑 divided by Δ𝑡.
We can also write this fraction in another way. If between a time of 𝑡 one and a time of 𝑡 two the object moves from a distance of 𝑑 one to a distance of 𝑑 two, then the object’s average speed 𝑉 is equal to 𝑑 two minus 𝑑 one divided by 𝑡 two minus 𝑡 one. Now, since a distance–time graph plots distance on the vertical axis against time on the horizontal axis, then if 𝑡 one, 𝑑 one and 𝑡 two, 𝑑 two are the coordinates of two points along a straight line drawn on a distance–time graph, that means that this expression for the speed 𝑉 is equal to the change in the vertical coordinate between these two points divided by the change in the horizontal coordinate between the same two points. In other words, this expression is equal to the slope of a straight line drawn on a distance–time graph.
We can say then that the speed of an object is equal to the slope of the corresponding line on a distance–time graph. A straight line is a line that has the same steepness at all points on that line. In other words, it’s a line with a constant slope. Then, since for a distance–time graph the slope of a line tells us the corresponding speed of an object, a straight line on a distance–time graph must represent an object that’s moving with a constant speed. We can see that the four lines on this distance–time graph are all straight lines. And this means that these four speed values will all be constant. For motion at a constant speed, the average speed of the object is the same as the speed at any point during that motion. So this expression for the average speed will give us the speeds for each of these four lines.
To work out each of these four speeds then, we need to pick two points on each line with coordinates 𝑡 one, 𝑑 one and 𝑡 two, 𝑑 two. Then, in each case, we can use these time and distance values in this expression in order to calculate the speed. Let’s now clear ourselves some space so that we can do this. We can notice that all four of the lines on this graph pass through the origin. That’s a time value of zero seconds and a distance value of zero meters. We can use the origin as the first point on all four of these lines. So, in all four cases, we have 𝑡 one equal to zero seconds and 𝑑 one equal to zero meters.
To choose the second point on each line, we can notice that at a time value of eight seconds, each of the four lines not only intersects with the vertical eight-second grid line, but they each intersect with a horizontal grid line as well. So let’s choose this point on the blue line, this point on the red line, this point on the green line, and this point on the orange line. All four of these points occur at a time of eight seconds. And so this is the value of 𝑡 two for all four lines. That means that with the points that we’ve chosen on each line, the only quantity that differs between the four lines is 𝑑 two. That’s the distance moved at the second point. Let’s label this second distance as 𝑑 subscript two b for the blue line, 𝑑 subscript two r for the red line, 𝑑 subscript two g for the green line, and 𝑑 subscript two o for the orange line.
Starting with the blue line and tracing horizontally across from the second point to the distance axis, we see that the distance 𝑑 subscript two b is equal to eight meters. Then, tracing across from the second point on the red line, we meet the distance axis at a height of six meters. So that’s our value for the quantity 𝑑 subscript two r. Doing the same thing for the second point on the green line, we find a value of four meters for 𝑑 subscript two g. Lastly, for the orange line, we get two meters for our value of 𝑑 subscript two o. Now, for each of the lines on the graph, we want to take our values for 𝑡 one, 𝑑 one, and 𝑡 two along with the particular value of 𝑑 two for that line and substitute them into this equation to calculate the corresponding speed.
Let’s begin with the blue line, so that’s calculating the speed 𝑉 subscript b. Based on this general equation, we know that 𝑉 subscript b is equal to 𝑑 subscript two b minus 𝑑 one divided by 𝑡 two minus 𝑡 one. Then, substituting in that 𝑑 subscript two b is eight meters, 𝑑 one is zero meters, 𝑡 two is eight seconds, and 𝑡 one is zero seconds, we get this expression for the speed 𝑉 subscript b. In the numerator, we’ve got eight meters minus zero meters, which is just the same as eight meters. And in the denominator, we have eight seconds minus zero seconds, which is just eight seconds. We have then that 𝑉 subscript b is equal to eight meters divided by eight seconds. This works out as a speed of one meter per second.
Now, let’s move on and do the same thing for the red line, so that’s finding the speed 𝑉 subscript r. The expression for the speed 𝑉 subscript r is going to look almost exactly like this one for 𝑉 subscript b. The only difference is that instead of this term 𝑑 subscript two b, for the speed 𝑉 subscript r, we have the quantity 𝑑 subscript two r. Substituting in that 𝑑 subscript two r is equal to six meters along with the same other three values that we had before, we get this expression here for the speed 𝑉 subscript r. In the numerator, six meters minus zero meters is six meters. And in the denominator, eight seconds minus zero seconds is eight seconds. So we have that 𝑉 subscript r is equal to six meters divided by eight seconds, which works out as 0.75 meters per second.
Now let’s do the same thing for the green line, so that’s finding the speed 𝑉 subscript g. This time in our expression for the speed along with the same three quantities 𝑡 one, 𝑑 one, and 𝑡 two, we’ve got the quantity 𝑑 subscript two g. Then, substituting in our values, we get this numerical expression for 𝑉 subscript g. We can then evaluate this to get a result of 0.5 meters per second. Our last speed left to find is 𝑉 subscript o, the speed corresponding to the orange line. This is equal to 𝑑 subscript two o minus 𝑑 one divided by 𝑡 two minus 𝑡 one. Substituting in our values for these four quantities gives us this expression for 𝑉 subscript o. This works out as a speed of 0.25 meters per second.
Okay, so we’ve now found the four speeds corresponding to the four different lines on this distance–time graph. Let’s clear some space on the board so that we can use these speed values to calculate these differences between the speeds of adjacent lines. The first difference is 𝑉 subscript b minus 𝑉 subscript r. Substituting in that 𝑉 subscript b is one meter per second and 𝑉 subscript r is 0.75 meters per second, this difference works out as 0.25 meters per second. Our second difference is 𝑉 subscript r minus 𝑉 subscript g. Substituting our values of 0.75 meters per second and 0.5 meters per second, this difference works out as 0.25 meters per second.
Then, the final difference is 𝑉 subscript g minus 𝑉 subscript o. Substituting in the values of 0.5 meters per second and 0.25 meters per second, this again works out as a difference of 0.25 meters per second. So we found then that all three of these differences are the same and are equal to 0.25 meters per second. Since we found the same difference between the speeds corresponding to each pair of adjacent lines on the graph, then our answer to this question is yes. The speeds corresponding to the lines on this distance–time graph do change value equally for any two adjacent lines.
