World Class Sprints

Enter the following command to load the necessary package.

<< Graphics`Graphics`

Since Mathematica already defines E as Euler's number, we do not need to make a separate definition.

Part 1. A First-Order Linear Model

Answer the questions in Part 1 here.  If you need to do any calculation, do so in the cell below.

Part 2. Acceleration, Velocity, Distance

Complete and enter the following command to define the function v that solves the differential equation

			(dv/dt) = A - (v/b),

and that satisfies the appropriate initial condition.

v[t_] :=  ?;

Find and enter the acceleration function a[t].

a[t_] :=  ?;

Find and enter the distance function s[t].  Be sure your function satisfies the appropriate initial condition.

s[t_] :=  ?;

Enter the following commands to plot the acceleration, velocity, and distance functions with 1973 values for the parameters.

A  =  12.2;
b  =  0.892; 
aplot  =  
      Plot[a[t], {t, 0, 10}, PlotStyle -> RGBColor[1, 0, 0],  
        DisplayFunction -> Identity]; 
vplot  =  
      Plot[v[t], {t, 0, 10}, PlotStyle -> RGBColor[0, 0, 1],  
        DisplayFunction -> Identity]; 
Show[aplot,  vplot,  
        DisplayFunction -> $DisplayFunction];
splot  =  Plot[s[t], {t, 0, 10}];

Answer the remaining questions in Part 2 in the text cell below.  Use the following Mathematica input cell for any additional calculations.

Answer:

Part 3. Parameter Values

Calculate the average split times for men.  (We will take up the women's data later.)

T1  =  ?;
T2  =  ?;
T3  =  ?;
T4  =  ?;

D1  =  30;
D2  =  60;
D3  =  80;
D4  =  100;

The following command will solve for b, given your choice of pairs (T1, D1) and (T2, D2).  You may copy and edit this command to solve for b with other pairs.

FindRoot[
      D1/D2 == (T1 - b + b*Exp[-T1/b])/(T2 - b + 
              b*Exp[-T2/b]), {b, 1}]

Enter your final value of b here.

b  =  ?;

Calculate A corresponding to your choice of b.  You may copy and edit this command to calculate A for other (time, distance) pairs.

A  =  D1/((b*(T1 - b + b*Exp[-T1/b])))

Enter your final value of A here.

A  =  ?;

Use the following commands to plot your model function and the sample data.  Change the data to your choice of runner or average runner.

mensModel  =  
        Plot[s[t], {t, 0, 10},  
          DisplayFunction -> Identity];
mensData  =  
        ListPlot[{{3.85, 30}, {6.45, 60}, {8.15, 80}, {9.87, 100}},  
          AxesLabel -> {"x"},   PlotStyle -> 
            RGBColor[1, 0, 0],  Prolog -> {PointSize[0.015]},  
          DisplayFunction -> Identity];
Show[mensModel, mensData,  
        DisplayFunction -> $DisplayFunction];

Copy and paste input lines for the men's model in the following input cell, and then edit to get the corresponding model for women sprinters.

Part 4. Summary

Answer the questions in Part 4 here:

World Class Sprints

Enter the following command to load the necessary package.

<< Graphics`Graphics`

Since Mathematica already defines E as Euler's number, we do not need to make a separate definition.

Part 1. A First-Order Linear Model

Answer the questions in Part 1 here.  If you need to do any calculation, do so in the cell below.

Part 2. Acceleration, Velocity, Distance

Complete and enter the following command to define the function v that solves the differential equation

			(dv/dt) = A - (v/b),

and that satisfies the appropriate initial condition.

v[t_] :=  ?;

Find and enter the acceleration function a[t].

a[t_] :=  ?;

Find and enter the distance function s[t].  Be sure your function satisfies the appropriate initial condition.

s[t_] :=  ?;

Enter the following commands to plot the acceleration, velocity, and distance functions with 1973 values for the parameters.

A  =  12.2;
b  =  0.892; 
aplot  =  
      Plot[a[t], {t, 0, 10}, PlotStyle -> RGBColor[1, 0, 0],  
        DisplayFunction -> Identity]; 
vplot  =  
      Plot[v[t], {t, 0, 10}, PlotStyle -> RGBColor[0, 0, 1],  
        DisplayFunction -> Identity]; 
Show[aplot,  vplot,  
        DisplayFunction -> $DisplayFunction];
splot  =  Plot[s[t], {t, 0, 10}];

Answer the remaining questions in Part 2 in the text cell below.  Use the following Mathematica input cell for any additional calculations.

Answer:

Part 3. Parameter Values

Calculate the average split times for men.  (We will take up the women's data later.)

T1  =  ?;
T2  =  ?;
T3  =  ?;
T4  =  ?;

D1  =  30;
D2  =  60;
D3  =  80;
D4  =  100;

The following command will solve for b, given your choice of pairs (T1, D1) and (T2, D2).  You may copy and edit this command to solve for b with other pairs.

FindRoot[
      D1/D2 == (T1 - b + b*Exp[-T1/b])/(T2 - b + 
              b*Exp[-T2/b]), {b, 1}]

Enter your final value of b here.

b  =  ?;

Calculate A corresponding to your choice of b.  You may copy and edit this command to calculate A for other (time, distance) pairs.

A  =  D1/((b*(T1 - b + b*Exp[-T1/b])))

Enter your final value of A here.

A  =  ?;

Use the following commands to plot your model function and the sample data.  Change the data to your choice of runner or average runner.

mensModel  =  
        Plot[s[t], {t, 0, 10},  
          DisplayFunction -> Identity];
mensData  =  
        ListPlot[{{3.85, 30}, {6.45, 60}, {8.15, 80}, {9.87, 100}},  
          AxesLabel -> {"x"},   PlotStyle -> 
            RGBColor[1, 0, 0],  Prolog -> {PointSize[0.015]},  
          DisplayFunction -> Identity];
Show[mensModel, mensData,  
        DisplayFunction -> $DisplayFunction];

Copy and paste input lines for the men's model in the following input cell, and then edit to get the corresponding model for women sprinters.

Part 4. Summary

Answer the questions in Part 4 here: