Here we will discuss speed and distance and time and graphing speed 


Speed 

Aamir and Jake are two best friends who want to see which one is faster. How can we determine who is the faster friend? What could we measure?

We could have two starting positions, position A, and position B, and both of them can start from position A  and run to position B. We as bystanders can measure the amount of time each person takes to run from position A to position B. Whoever has the shortest time would definitely be the faster friend. Another type of measurement we can measure is also distance. Distance is usually measured in meter. So let us say that position A is 20 meters away from position B. What we now have to do is check for the shortest time.

Let's say that it took Aamir 20 seconds to run 10 meters, and it took Jake 15 seconds. We can see that Jake is faster (because he had the shorter time).  

But what if Jake ran a shorter distance, like 10 meters, and Aamir ran 20 meters. How can we determine which one is faster. How can we possibly compare the 20 meters to 10 meters?

Response:

  1. Jake would be faster because it does not matter who ran for a longer distance, we only care who took the shorter time.

  2. Aamir would be faster because we only care about who ran a longer distance.

  3. We don not know because not enough information is given.

  4. We need to see how much distance they can cover in 1 second.

The correct answer is 4. This is because even if we look at time, or distance individually, we do not get an accurate answer as to who is faster.

The way in which we determine who is faster is by comparing two things. In math, when we compare something, we look at their ratios. In our problem, what should we compare to see which person is faster:

Responses:

  1. We look at Aamir’s time and compare it to Jake’s time

  2. We look at Aamir’s distance and compare it to Jake’s distance

  3. We look at each person individually and compare their distance to their time

The correct answer is 3. In order to compare two things we have a very power mathematical operation we can use, division. With division, we can compare the distance to time.

Let us do that. We will look at Aamir first. Aamir’s distance is 10 meters, and his time is 20 seconds. So, now we divide 10 meter by 20 seconds.

10 / 20 = 0.5

We get 0.5. What does this number mean? Well we are comparing meters to seconds, so our answer would be meters over second (or we could cay meter per second). This is called speed, so we should give it the symbol is s. This means that when Aamir runs 0.5 meters, 1 second would have passed.

Now let us look at Jake’s speed. Jake ran 10 meters and it took him 15 seconds. What is his speed?

Response: 

  1. 10/15

  2. 15/10

The correct answer is 1 which is: 10/15, or 0.667 meters/second. Now that we have their speed, can we determine who is faster, well yes but how? Well we, just compare 0.667 to 0.5.

0.667 > 0.5

Jake’s speed is bigger than Aamir’s speed so we say that Jake is faster.

Working with just numbers could be a bit tedious, so we need to find a way to visualise this information. We can use a graph. In graph we have an x-axis and a y-axis, and a point on a graph is a coordinate. What would be our x-axis and our y-axis?

Response: 

  1. x-axis is speed, y-axis is time

  2. x-axis is time, y-axis is distance

  3. x-axis is distance, y-axis is time

The answer is 2. The x-axis is time and the y-axis is distance. We could have had 3 as our answer, but the we usually have time on the x-axis. This is because time is  an independent variable, and distance is a dependent variable. An independent variable is a variable that does not change because of another variable; however, a dependent variable change because another variable changed. In our problem, the time increased and because of that our distance also increased.

With graphs we need to put points on our graphs. We know that our x-axis has time and y-axis has distance, so that is what we will put on our graph.

We know that Aamir at 0 seconds ran 0 meters, so that is one point.

Amir's initial position: (0, 0)

Also at 20 seconds, he finished his run and was 10 meters away from his original position, so our coordinate is

Amir's final position: (20, 10)

now we can look at the graph.

 

Graph of Time and Distance for Amir's Speed

As you can see, we have two points, we can connect them to get a line.

The line, mathematically has specific properties. For one, lines are not curved. They are straight. If the line has two end points, like the one we have, it is called a line segment. This means that the line does not go on forever. The two points of a line are always fixed. Also notice that we only needed two points to make a line. This is also another important characteristic.

Lastly, the most important characteristic of a line is the slope. The slope is  mathematically defined as the change in y divided by the change in x. Also, when I say “mathematically defined,” by that I am referring to specific properties that we as humans have come up with to understand something. Here we are using the slope to understand a line.

We use the Greek letter delta to mean change in. When we say change in time we mean, the final time minus the initial time,

Delta t = t_final - t_initial

We can also attach the delta to distance as well,

Delta d = d_final - d_initial

So the slope is written like this,

Slope = delta y / delta x

In our problem, our y coordinate is distance so our, delta y is delta d, and our x coordinate is time, so our delta x is delta t.

Slope = delta y / delta x = delta d / delta t

The above equation is the same as,

Slope = d_final - d_initial / t_final - t_initial

 

What is our d_final, d_initial, t_final, and t_initial?

Response: 

  1. D_final = 0, d_initial = 10, t_final = 10, t_initial = 0

  2. D_final = 10, d_initial = 0, t_final = 20, t_inital = 0

  3. D_final = 20, d_initial = 0, t_final = 10, t_initial = 0

The answer is 2. We can look at the graph to get our values. Now we do the arithmetic,

Delta d = d_final - d_initial = 10 - 0 = 10

Delta t = t_final - t_initial = 20 - 0 = 20

Here is why you may wonder, why did we have to subtract the initial numbers, they are zero?

The reason we did that is because if our numbers were different and not zero then when we calculated the slope our slope would have been wrong. Imagine the line as the hypotenuse of a right angle triangle. The time is the first leg, and the distance is the second leg. We want to see how they change.

We calculate the slope now,

Slope = delta d / delta t = 10 / 20  = 0.5

 

What is interesting about our slope is that it is Aamir’s speed. Now we can generalize this information. The slope of a distance time graph is the speed.

 

Now, draw a graph for Jake’s speed and calculate the slope.