This module illustrates a modeling process and a somewhat unusual application of first-order linear equations. It is appropriate for both calculus courses and differential equations courses -- wherever these equations are taken up.

David Smith is Associate Professor Emeritus of Mathematics at Duke University.

Click on the button corresponding to your preferred computer algebra system (CAS). This will download a file which you may open with your CAS. You should arrange your browser and CAS windows side by side so you can see both.

Ver. 6 or higher |
Ver. 3.0 or higher |
Ver. 5.1 |

This module was prepared with the assistance of Jer-Chin Chuang (Furman University) and John Michel (Marietta College). It is based on an article by S. R. Dunbar in the *CODEE Newsletter*, Spring,1994.

This module illustrates a modeling process and a somewhat unusual application of first-order linear equations. It is appropriate for both calculus courses and differential equations courses -- wherever these equations are taken up.

David Smith is Associate Professor Emeritus of Mathematics at Duke University.

Click on the button corresponding to your preferred computer algebra system (CAS). This will download a file which you may open with your CAS. You should arrange your browser and CAS windows side by side so you can see both.

Ver. 6 or higher |
Ver. 3.0 or higher |
Ver. 5.1 |

This module was prepared with the assistance of Jer-Chin Chuang (Furman University) and John Michel (Marietta College). It is based on an article by S. R. Dunbar in the *CODEE Newsletter*, Spring,1994.

In 1973, J. B. Keller published a theory of competitive running, in which he proposed that the speed of a sprinter (up to 300 meters) could be modeled by the differential equation

**dv/dt = A - v/b**,

where **v(t)** is the speed at time **t**. At the time of Keller's work, reasonable values for the constants were **A = 12.2** m/sec^{2} and **b = 0.892** sec.

- What is an appropriate initial condition for the model?
- Solve the initial value problem symbolically, using the parameters
**A**and**b**-- don't substitute numbers yet. You should be able to do this step with pencil and paper, but use your helper application if you need to. [You may have encountered problems of this type in many different contexts: RL circuits, Newton's Law of Cooling, exponential growth and decay, velocity in a resisting medium, and mixing problems.] - What is the limiting behavior of
**v**as**t**becomes large? What meanings can you attach to the parameters**A**and**b**?

- Find the acceleration function
**a(t)**as an explicit function of time**t**, still in terms of the parameters**A**and**b**. - Find the distance function
**s(t)**as an explicit function of time**t**, still in terms of the parameters**A**and**b**. - Plot the acceleration, velocity, and distance functions, using the 1973 values of the parameters
**A**and**b**. - When does the
*maximum acceleration*occur, and what is it? How long does it take for the acceleration to drop to 10% of its maximum value? - What is the runner's
*maximum speed*? How long does it take for the runner to reach 90% of this maximum? In races of 100 and 200 meters, is the*final speed*the same? Explain.

We turn now to the question of appropriate values for the parameters 20 years after Keller's work. The following tables record 1993 world-class performances in the 100 meter sprint, separately for men and women. Each table shows "split times" for 30, 60, 80, and 100 meters.

1993 World Championships

Name |
30 m |
60 m |
80 m |
100 m |

Linford Christie | 3.85 | 6.45 | 8.15 | 9.87 |

Andre Carson | 3.83 | 6.43 | 8.15 | 9.92 |

Dennis Mitchell | 3.82 | 6.46 | 8.22 | 9.99 |

Carl Lewis | 3.95 | 6.59 | 8.30 | 10.02 |

1993 World Championships

Name |
30 m |
60 m |
80 m |
100 m |

Gail Devers | 4.09 | 6.95 | 8.86 | 10.82 |

Merlene Ottey | 4.13 | 6.98 | 8.87 | 10.82 |

Gwen Torrence | 4.14 | 7.00 | 8.92 | 10.89 |

Irina Privalova | 4.09 | 7.00 | 8.96 | 10.96 |

To illustrate the meanings of split times -- and the possibility that Keller's parameters are outdated -- we show the men's split times in the following figure, along with possible model distance functions. Christie's data (fastest) are shown as solid circles, and Lewis's (slowest) as open diamonds. The broken curve shows the distance function you calculated in Part 2 with Keller's parameters. The solid curve shows a possibly better fit -- with a faster terminal velocity.

- Calculate
*average*split times for*men*for each of the distances. (We will take up the women's data later.) We call these times**T**, corresponding respectively to the distances_{1}, T_{2}, T_{3}, T_{4}**D**_{1}= 30, D_{2 }= 60, D_{3}= 80, D_{4}= 100.

- Each
**(time,distance)**pair gives you numbers to substitute into your distance formula to get equations that involve only**A**and**b**. If you take ratios of two distances, say**D**and_{1}**D**, you will have an equation that involves only_{2}**b**. Why? If the corresponding times are**T**and_{1}**T**, explain why the resulting equation is_{2}

Before we charge ahead to find a value of **b** from the displayed equation, we make a few observations about this equation and its relation to the modeling process.

- When we have two parameters to determine from the data, such as
**A**and**b**, there is an advantage in having an equation that involves only one of them. It is much easier to solve equations in one unknown than in two. - Once we have a reasonable value for
**b**, we can use that in the time-distance relation to find a corresponding value of**A**. - Unless our model is a
*perfect*fit to the data -- which hardly ever happens -- we will get a*different*determination of**b**from each pair of (time,distance) pairs. If these values are reasonably close to each other (i.e., consistent), it may make sense to average them. If not, we should rethink whether the model is really appropriate for the data. - We obtained the displayed equation by taking the ratio of
**D**to_{1}**D**-- but we would get the same information about_{2}**b**if we had taken the ratio of**D**to_{2}**D**. That is, the same value of_{1}**b**would satisfy both equations. Thus, there is no point in taking ratios both ways. - Given four
**(time,distance)**pairs to choose from,**(T**_{1},D_{1}), (T_{2},D_{2}), (T_{3},D_{3}), (T_{4},D_{4}),there are six different ways to pair them up so the first subscript is smaller than the second: (1,2), (1,3), (1,4), (2,3), (2,4), (3,4). Thus, we get six possibly different determinations of

**b**from our men's data.

Before we tackle the problem of finding a numerical value for **b**, we will look at the problem graphically to see if it makes sense to proceed.

- Plot the difference between left-hand and right-hand sides of the equation displayed in step 2, as a function of
**b**, to isolate and estimate a value for**b**. - Copy and edit your plotting command to produce corresponding plots for the other five combinations of subscripts. Plot all six graphs together to see if they give relatively consistent information about
**b**. Record your best estimate of**b**from the visual information. - Solve the equation in step 2 for
**b**, using the first two**(time,distance)**pairs for the men. (The solution may take a little time -- be patient.) Then vary the**(time,distance)**pairs several times to get your best estimate of the parameter**b**for the men. - Explain why
**A**can now be found from the equation

- Solve for the parameter
**A**for the men, again using several choices of**(time,distance)**pair to get your best estimate.

- As a check on your work, plot your distance function for men together with the data for one of the men (or the average data, if you prefer).

- Repeat steps 1, 3, 4, 5, and 7 to estimate the parameters
**b**and**A**for women.

- Repeat step 8 with the women's distance function and data.

- Describe in your own words the extent to which you do or don't find the first-order linear model a reasonable representation of a sprinter's speed.
- What is the significance of the apparent change of parameters in the model over the time span from 1973 to 1993?

- What is the significance of the difference between parameter values for men and women?
- Some track experts speculate that men's and women's abilities in the sprints are becoming identical. Do your conclusions support this? Why or why not? Do you have enough evidence to determine whether there has been a change in relative abilities since 1973? Explain.