5. Sprinting – Effects of the
Athlete’s Muscle Power
OBJECTIVES
•
Use sophisticated software (Mathematica) to calculate the velocity profile of an athlete in a
60-m sprint.
•
Examine how the athlete’s maximum anaerobic muscle power affects the velocity profile and
sprint time.
INTRODUCTION
In this model the energy required for sprinting is released by chemical conversion in the athlete’s
muscles. The energy passes through a variety of intermediate stages (including the mechanical
work of the muscles, elastic energy in the muscles and tendons, and kinetic and potential energy of
the limbs and centre of mass), but is ultimately degraded to heat or is accounted for by the external
work expended on the centre of mass of the athlete (Arsac & Locatelli, 2002; Linthorne, 2013).
The power balance equation for the athlete is
η (Pmuscle – Pheat) = Pext
(1)
where Pmuscle is the power released by chemical conversion in the athlete’s muscles, Pheat is the
mechanical power that is degraded to heat, Pext is the external power expended on the centre of
mass of the athlete, and η is the efficiency of converting metabolic energy to external work.
The power generated by chemical conversion in the athlete’s muscles is the sum of the
contributions from anaerobic and aerobic metabolism:
Pmuscle = Panamax exp(–t/τana) + Paermax [1 – exp(–t/τaer)]
(2)
where Panamax is the maximum power available from anaerobic metabolism, Paermax is the maximum
power available from aerobic metabolism, τana is the time constant for the release of anaerobic
energy, and τaer is the time constant for the release of aerobic energy. Equation 2 expresses the
finding that in running the power released by anaerobic metabolism decreases rapidly with time
from an initial maximum value and that the power released by aerobic metabolism slowly increases
with time towards a maximum value. In a sprint over 200 m or less the power generated by the
athlete’s muscles is mostly produced through anaerobic metabolism.
5000
power (W)
4000
total muscle power, Pmuscle
3000
anaerobic power
2000
1000
aerobic power
0
0
2
4
6
1
8
time (s)
10
12
14
Sprinting Lab
A large fraction of the power generated by the athlete’s muscles is eventually dissipated as heat. In
sprinting the mechanical power degraded into heat is proportional to the athlete’s running velocity,
v, and so can be expressed as
Pheat = A v
where A is the rate of degradation of mechanical energy into heat.
(3)
The rate of performing external work on the centre of mass of the athlete is given by
Pext = Pinertia + Pdrag
(4)
where Pinertia is the power expended in overcoming the inertia of the athlete and Pdrag is the power
expended in overcoming aerodynamic drag. The kinetic energy of the athlete is given by KE =
½mv2, where m is the mass of the athlete. The rate of change of the kinetic energy of the athlete
(i.e., the power expended in overcoming inertia, Pinertia) is then d(KE)/dt = mv(dv/dt). The power
expended in overcoming aerodynamic drag is given by the product of the drag force (Fdrag) and the
velocity of the athlete (v). The aerodynamic drag force acting on the athlete is given by
(5)
Fdrag = ½ ρ S CD v2
where ρ is the air density (1.225 kg/m3 at sea level and 15ºC), S is the cross-sectional area of the
athlete (0.50 m2), CD is the drag coefficient (0.7), and v is the speed of the athlete relative to the air
(up to about 12 m/s). Equation 4 then becomes
dv
Pext = m v  dt  + ½ ρ S CD v2 v .
 
(6)
The power required to perform an athletic movement is affected by the efficiency of converting
metabolic energy to external work. At the start of a sprint the efficiency of running is assumed to
be about η = 0.25, which is the usual efficiency of a concentric muscle contraction. In this model
the running efficiency increases linearly with running velocity as the sprinting action becomes
enhanced by storage and recoil of elastic energy in the athlete’s muscles and tendons, eventually
reaching a value of η = 0.5 when sprinting at maximum velocity. Therefore, the efficiency of
converting metabolic energy to external work during a sprint is given by
0.25 v
η = 0.25 + v
max
where vmax is the athlete’s maximum unloaded running velocity.
(7)
Equation of Motion of the Athlete
Combining equations 1–7 gives

dx
0.25  dt 

  
dx
0.25 +
 Panamax exp(–t/τana) + Paermax [1 – exp(–t/τaer)] – A   
vmax  
 dt  

2
3
dx d x
dx
= m  dt   dt2  + ½ ρ S CD  dt 
(8)
  
 
2
2
where v = dx/dt and a = d x/dt are the first (velocity) and second (acceleration) derivatives of
position (x) with respect to time (t). Equation 8 is the ‘equation of motion’ that describes the
behaviour of the athlete as a function of time. The equation is a second-order differential equation
in time. Because it is non-linear it must be computed using numerical methods. Here, we will
obtain a numerical solution for x(t) using a technical computing software package (Mathematica).
The table below shows typical values of variables for elite sprinters. These values were obtained by
analysing the velocity-time curves of athletes in the 100-m finals at recent Olympic Games.
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Sprinting Lab
Variable
Body mass, m (kg)
Maximum anaerobic power, Panamax/m (W/kg)
Maximum aerobic power, Paermax/m (W/kg)
Men
Women
80
60
90 – 80
77 – 70
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We also use the following values for a ‘standard’ athlete: A/m = 4.0 W/kg per m/s for the rate of
degradation of mechanical energy into heat; τana = 13 s and τaer = 25 s for the rates of release of
energy for anaerobic metabolism and aerobic metabolism; and vmax = 10 m/s for the athlete’s
maximum unloaded running velocity.
PROCEDURE
Go to one of the University computers and log on to the computer in the usual way. Open My
Computer, then double-click on the L Drive (depapps on ‘academic.windsor’ (L:)). Open the
CC folder, then open the Math8 folder. Start up Mathematica by double-clicking the
Mathematica icon.
Open a web browser (Netscape or Internet Explorer) and go to the “Laboratory Items” page on the
Biomechanics of Sport and Exercise Blackboard site. Sprint.nb is a Mathematica notebook with a
program that calculates the trajectory of a sprinter under the influence of aerodynamic drag. Click
on the icon and download the file from the Blackboard site to your computer hard disk. (I suggest
you place the files in the My Documents folder.)
Return to the Mathematica window and select Open from the File menu. The Open dialog box
will appear. Find and open the Sprint notebook file that you downloaded to your computer.
The Sprint program calculates the position, velocity, and acceleration of an athlete in a 60-m sprint.
The sprinter trajectory is calculated with a ‘vacuum model’ (solution1) in which air resistance is
ignored, and an ‘aerodynamic model’ (solution2) which includes the effects of aerodynamic drag.
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Sprinting Lab
Understanding the Program
Have a careful look at the program and try to make sense of it. The plots at the bottom of the
program listing contain the output of the computer program. The red curve is for the vacuum model
and the blue curve is for the aerodynamic model.
60
distance HmL
50
40
30
20
10
0
2
4
time HsL
6
8
0
2
4
time HsL
6
8
12
velocity HmêsL
10
8
6
4
2
Mathematica uses the Lagrange notation to indicate differentiation, where x′ is the first derivative
and x″ is the second derivative. The equation of motion of the athlete (equation 8) is therefore
written as
0.25 x′

0.25 + v
 (Panamax exp(–t/τana) + Paermax [1 – exp(–t/τaer)] – A x′)

max 
3
= m x′ x″ + ½ ρ S CD (x′) .
(9)
In the Mathematica program, this differential equation is solved numerically using the NDSolve
function. The following line of code
0.25 x′[t]

solution2 = NDSolve[{0.25 + Vmax  (Panamax Exp[–t/Tana] + Paermax (1 – Exp[–t/Taer])


– A x'[t]) == m x′[t] x″[t] + ½ Density S Cd x'[t]3, x[0] == 0, x'[0] == 0.0001}, {x}, {t, 0, Tmax}]
finds a numerical solution for equation 9 and assigns it to the variable solution2. The segment of
code {x},{t, 0, Tmax} means that the equation is solved for x, with t in the range from 0 to Tmax.
The segment of code x[0] == 0, x'[0] == 0.0001 assigns the initial position and velocity of the
athlete (at t = 0) to 0.
The results are plotted using the Plot and ParametricPlot functions. The following line of code
Plot[{Evaluate[ x[t] /. solution1 ], Evaluate[ x[t] /. solution2 ]}, {t, 0, Tg}, PlotRange → {0, 60},
Frame → True, FrameLabel → {“time (s)”, “distance (m)”}, PlotStyle → {Hue[0.05], Hue[0.7]}]
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Sprinting Lab
produces a displacement-time graph of the solutions to the vacuum and aerodynamic models. The
segment of code, Evaluate[ x[t] /. solution1], generates the x values for the vacuum model, and the
segment of code {t, 0, Tg} indicates that these x values will be plotted as a function of t, with t
ranging from 0 to Tg. The range of values on the vertical axis is specified using PlotRange, and
the axis labels are inserted using FrameLabel. A frame is drawn around the plot using Frame.
The colours of the two curves are assigned using Hue. Colours can vary anywhere along the
rainbow (red, orange, yellow, green, blue, indigo, violet, and back to red again) by assigning a
value of between 0 and 1.
To produce plots as a function of velocity (rather than time) we use the ParametricPlot function.
The following code
ParametricPlot[{Evaluate[ x[t], x'[t] /. solution1 ], Evaluate[ x[t], x'[t] /. solution2 ]}, {t, 0, Tg},
PlotRange → {{0,60},{0,12}}, Frame → True, FrameLabel → {“distance (m)”, “velocity (m/s)”},
PlotStyle → {Hue[0.05], Hue[0.7]}]
produces a velocity-displacement graph (i.e., x'[t] versus x[t]) of the solutions to the vacuum and
aerodynamic models.
Test Runs
The plots shown above are with a maximum anaerobic power of 70 W/kg. You can change the
athlete’s maximum anaerobic muscle power by typing in new values for Panamax. To run the
Sprint program with the new values, select Evaluation from the Kernel menu, then select
Evaluate Cells. Experiment with different anaerobic power values and note the changes in the
curves. (You could also experiment with changing the values of some of the other variables and
noting the changes in the curves.)
Effect of the Athlete’s Maximum Anaerobic Muscle Power
Here, you will systematically investigate the effect of changes in the athlete’s maximum anaerobic
power (Panamax/m) on their velocity profile and sprint time. An athlete who has a greater power-toweight ratio is expected to produce a more rapid acceleration and reach a faster top speed.
Open a web browser (Netscape or Internet Explorer) and go to the “Laboratory Items” page on the
Biomechanics of Sport and Exercise Blackboard site. SprintPower.nb is a Mathematica notebook
with a program that calculates the trajectories of sprinters with different power-to-weight ratios.
Click on the icon and download the file from the Blackboard site to your computer hard disk.
Return to the Mathematica window and select Open from the File menu. Find and open the
SprintPower notebook file.
The program SprintPower produces graphs with curves for only 3 different values of Panamax/m.
Modify the program so that it displays several more curves (i.e., 5–7 in total) in the graphs. You
should obtain a graph similar to the one shown below.
5
Sprinting Lab
12
velocity HmêsL
10
8
6
4
2
10
20
30
distance
HmL
40
50
60
Exporting Data to Excel
If you wish, you can export the data that was used to produce the Mathematica graphs into an Excel
spreadsheet and then re-draw the graphs. To export the Mathematica data, insert a line of code
similar to the following in your program;
Export["d:/datafileSprint.csv", Table[{t, x[t]/.solution1, x'[t]/.solution1, x''[t]/.solution1},
{t, 0, 7, 0.1}], "CSV"]
In this example, the data will exported into an Excel CSV file called datafileSprint on the D Drive.
The file will be a table of data of the time, position, velocity, and acceleration of the solution to the
equation specified in solution1. The line of code {t, 0, 7, 0.1} indicates that the time data will range
from 0 to 7 seconds in increments of 0.1 seconds.
After running the Mathematica program, open the Excel file that it generates to view the data. Note
that some of the columns of numbers in the spreadsheet include unwanted brackets, { }. However,
you can remove these brackets by using the Replace function. From the Find and Select menu on
the Home tab, select Replace. Insert a “{” in the Find what box, and nothing in the Replace with
box, then click the Replace All button. Repeat the process to remove the “}” brackets. You can
now plot the data as desired.
Other Investigations
You could also modify the Mathematica program to examine the effects of wind or altitude in
sprinting. It will require some careful thought to produce the appropriate equations and the changes
to the program.
In some instances the required changes to the equations are relatively minor and can be easily made
using the mouse and keyboard. However, if more substantial changes to the equation are required
you should use the input palette to help produce the equation. To do this, select BasicInput from
the Window menu. The input palette will appear. When you click on a button in the palette the
character or object shown in that button will be inserted into your program.
References
Arsac, L. M., & Locatelli, E. (2002). Modelling the energetics of 100-m running by using speed
curves of world champions. Journal of Applied Physiology, 92, 1781–1788.
Linthorne, N. P. (2013). A mathematical modelling study of an athlete’s sprint time when towing a
weighted sled. Sports Engineering, 16, 61–70.
Copyright © 2009–2013 Nick Linthorne
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