Question

The Olympic record for the men's 400-meter hurdle race is 45.94 seconds. It was set by Karsten Warholm in 2021. His average running speed was 400 = 45.94 28.71 meters per second (distance divided by time equals speed) 1. Make a table and a graph showing how 400 meter race time changes as average speed increases from 2 meters per second to 10 meters per second in steps of one meter per second (in other words, the speed column should go 2, 3, 4, --. 10). Also remember that distance is equal to rate (or speed) times time. d=yt 2. Describe the pattern of change shown in your table and graph 3. Write a rule (equation) showing how to calculate race time, t. for any average speed, s. 4. Which change in speed will reduce rate time the most an increase from 2 to 4 meters per second or an increase from 8 to 10 meters per second? Explain how your answer is illustrated in the shape of your graph.

Answer 1

We are given a distance of 400 meters and we are asked to determine the time to cover this distance for different values of speed. Since we are asked to construct a table, we begin by setting up a column with values of speed that range from 2 meters per second to 10 meters per second, like this:

Now we need to set up another column with the values for time. To do that we will use the fact that the speed is defined as:

`v=(d)/(t)`

Where "v" is speed, "t" is time, and "d" is distance. We can solve for the distance by multiplying both sides by "t":

`vt=d`

Now we divide both sides by "v":

`t=(d)/(v)`

Now, since the distance is fixed at 400 meters we can replace this value in the equation:

`t=(400)/(v)`

Now we have a relationship that will allow us to determine the value of the time. We just need to replace each of the values on the column for speed into the equation to get the value for the time. Replacing the first value of speed we get:

`t=\frac{400\text{ m}}{2\text{ m/s}}=200s`

This is the first value for the column of time. The second value is obtained replacing the second value for the speed:

`t=\frac{400\text{ m}}{3\text{ m/s}}=133.3s`

This is the second value. We continue like this for each of the values and we get the following table:

Now we can plot these values and we obtain the following graph:

3. The rule to calculate race time "t" for any average speed is the one that was determined previously to construct the table, this is:

`t=(400)/(s)`

4. we are asked about the change in time given a change in speed. We can calculate this by subtracting the initial time from the initial speed from the final time at the final speed, this is:

`\Delta t=t_(v2)-t_(v1)`

Where:

`\begin{gathered} \Delta t\text{ = change in time} \\ t_(v1)=\text{ time at velocity 1} \\ t_(v2)=\text{ time at velocity 2} \end{gathered}`

Now, the change from 2 to 4 meters per second is the following:

`\begin{gathered} t_2=200 \\ t_4=100 \end{gathered}`

replacing we get:

`\Delta t_(2-4)=100-200=-100`

Therefore, the time is reduced in 100 seconds. Now for an increase from 8 to 10 meters per second we get:

`\begin{gathered} t_8=50 \\ t_(10)=40 \end{gathered}`

Replacing in the formula for change:

`\Delta t_(8-10)=40-50=-10`

therefore, the time is reduced in 10 seconds. This means that the change from 2 to 4 meters per second will reduce the race time the most. This can be illustrated in the graph as it gets flatter the greater the speed is, meaning that the changes in time are reduced.