### Math 505to2

```The Effects of Wind and Altitude in the 400m Sprint with Various
IAAF Track Geometries
Vanessa Alday and Michael Frantz
• Results similar to those for standard track.
• Optimal wind direction for lane 4 is still in the 210°-240°
range, for lane 8 at 240°.
• Lane 1 remains disadvantaged at all wind angles.
• Best race times are achieved at angles of either around 120° or
240°.
Presented by: Crystal Bennett, Adrienne Godette, Shywanda Surgeon,
Constance Williams
Objective
Effects of Wind Direction
• Investigate the effects that wind and altitude have on the 400m
sprint when run on various IAAF track geometries.
• To validate Quinn’s ordinary differential equations model
using data from the 1999 World Athletics Championships.
• Maple is used to solve the model’s equations numerically.
• Find performance differences between running lanes,
indicating possible disadvantages of running in certain lanes.
This model further reveals that the runners total velocity in
relation to the radius of the bend is
where  satisfies

the cubic equation 3 + 2 − 2 = 0.  =
2 where R is the
• The ancient Greeks used an panathenaic stadium during their
Olympic games.
bend radius and g is the gravitational force.  is the runner’s peak
velocity.
• The track geometry is different with the 180 meter straights
and the 20 meter curves.

The cubic equations is satisfied by
• The panathenaian track permits longer runs at peek speed, but
the curves slow the runner down tremendously.
Background
Where  =

2
..
Computation of the Effects of Wind
Further considerations for the runners is how they are placed on the track.
It well understood that the runners are staggered around the track based
on their lane. Each runner runs a different length around the bend at
different intervals of the race.
The wind velocity  which is the wind velocity relative to the ground
and tangent to the path of the runner. This value will vary in each four
segments of the track and depends on wind velocity and relative wind
direction.
Although track design is standard, wind and altitude
vary
Models have been developed for the 100m, 200m and
4x100m relay races as far back as the late 1920’s. These
were relatively simple review.
A model for the 400m was not developed until 2004
and presents unique problems.
• Consideration for the runner covering the entire track
during the race.
• The path for each runner will vary because of the
range of radii of the track lanes.
• Research suggests wind blowing across a closed loop
track has a detrimental effect on a runner’s time.
Early Models

=  −
() is the runner’s velocity at time
() is the runner’s total propulsive force per unit mass
is the damping coefficient
1
τ

A more modern model known as Quinn’s model that extends Keller’s

=  − −

τ
The component of the wind blowing in the direction of the runner for a
runner traveling in a straight line with a wind velocity of blowing at an
angle  of is  =  cos() The angle is shown in the figure
following.
− ( −  )2
Details of the equation:
− is the propulsive force per unit mass that declines as the athlete’s
muscles tire.
ρ
=  where  is air density,  is the coefficient of drag, and  is the
2
frontal area of the athlete.
is the velocity of the wind.
The IAAF Track and its Effect On Runners
The IAAF track which is known as the standard running track is
400 meters long with two straights 84.4 meters, and two bends 115.6
meters long. So 57.8% of the 400m race is run around bends.
Runners have the same run along the straights but not along the
bends since they vary in radii. So runners will experience a
different wind velocity which will effect the velocity the velocity
the runner can maintain around the bend.
It is obvious that the velocity around the bend is less than that run
along the straights. The runner has to adjust the distribution of
their weight to compensate for their run around the bend.
Effects of Altitude and Air Density
•  is the drag coefficient.
• Altitude effects the aerobic
energy system
•  is the frontal area of athlete
• Runners favor high altitude
because of less air density
•  is the mass
ρ
• Modify expression  =  ,
2
replace ρ ,  is distance
above sea level in meters.
• So ρ = ρ0

(−273)
• The equal quadrant track differs from the standard track only
in its dimensions.
• So all the equations will remain the same, except for the wind
velocity.
• Changes to the wind velocity functions can be easily
determined, and the simulation run with virtually the same
code.
• Using the standard track data from Ward-Smith and Radford
(2002), but now for an equal quadrant track, we have the lane
Comparison of lane 4 (windless) times; IAAF standard vs. Panathenian track
Effect of a 2 m/s wind on lanes 1, 4, and 8, with time
corrections from windless (Panathenian track)
• T is air temperature (Celsius)
•  is air density.
• Gas Constant R,  = 287 ∙
−1 ∙  −1
Time correction factors for a 2 m/s wind on the
Panathenian track
The geometry of the IAAF standard track and wind
direction
Results
Relative wind velocity around the IAAF standard
track
Time corrections for lanes 1, 4, and 8 for different altitudes in
windless conditions
Altitude and Propulsive Force
• Considered the propulsive force,
One of the earliest recorded models for energy balance of a runner was
provided by Archibald V. Hill and is based on Newton’s Law.
Later a model is presented by J.B. Keller that expands Hill’s:
is shown:
Data
Runners performance is affected by different wind directions angles .
Effect of a 2 m/s wind on lanes 1, 4, and 8, with time corrections from windless
The International Associate of Athletics Federations track
(IAAF)
This track has eight lanes to accommodate a maximum
of eight runners.
Each track is 400 meters in length with each lane having
Details of the equation:
The two tracks
-
Propulsive force- propelling force that causes something to move
forward or onward.

• In model  = 1.184 3

• T = 25°C
• Altitude H, varies.
• Image contains time corrections for lanes 1, 4, and 8
on standard track at various altitudes under windless
conditions.
• Factor that affects the propulsive force:
- Aerobic energy system ( the muscle energy that requires oxygen)
- contributes significantly to the amount of energy
supply during a long sprints.
- total contribution for 400m sprint was found to be 43%
- Concludes that aerobic energy system is affect by the altitude.
• The faster rates in higher altitudes causes a decline in the
propulsive force.
• β depends on the altitude H and the aerobic energy
system
• Rewrote  =  (1 + )
•  - the parameter value at the sea level.
Results of Quinn model
Analyzing as before, we obtain in Table 7 new expressions on different portions of the
track for the relative wind velocity .
Wind Effects on the Equal Quadrant Track
Comparison of times for equal quadrant and
standard tracks (windless)
• Time corrections are almost identical across lanes at each
altitude.
• Expected, The only effect on race times due to lane geometry
comes from physical mechanics of negotiation of tighter
bends, which should not be affected by altitude.
• Largest time correction was 0.32 sec.
• Big deal for world records.
• Why it has been accepted races at altitudes higher than 1000
meters, these performances are marked by an “A” (IAAF)
• The biggest disadvantage to the runner in windless conditions
are in lane 1, since the tighter bends the runner spends more
time on the ground rather then in the air.
• Lane 8 is the fastest lane having shorter distance around the
bend the runner stays in the air longer.
• Wind directions parallel to the straights provide the worst
conditions.
• It is to the runner advantage to have the wind on its back first
earlier and run into the headwind later.
• Wind conditions have such an advantage on a runner that it
could change a winner and loser on different days.
• Lane selection, wind speed, wind direction, location ,
altitude, and even track geometry play an important role on a
400 meter sprint.
Future Work
• Quinn's 2004 model considered winds of
constant speed and direction throughout the
included in the wind off the stadium onto the
track.
• Measuring modeling and including factors such
as runners performance, physical conditions,
clothing, and nervous tension. Which are all
almost impossible.
• Strategy influences such as whether to beat the
other runners, the fastest time, or their personal
time. This is also non-quantifiable.
References
• Using previous data from a 400m sprint, where
• (maximum force) = 7.91m/s²
• ( decay rate) = .006 1/s
•  (resistive force) = 1.45s for men
• Compared a model written using Maple
Our model times closely approximate Quinn's results, the major difference being
0.04 seconds at 150m. The finishing time is the same for all three models.
•
• IAAF standard track is the norm for track dimensions
• IAAF accepts other tracks. Like equal quadrant track
• 400m track, 100m bends,100m straightaways, measured along
lane 1.
• Our model has only been applied to the IAAF standard track,
but we will now examine how the wind affects a runner's
performance on an equal quadrant track.
Effect of a 2 m/s wind on lanes 1, 4, and 8, with
time corrections from windless
Previous data smoothed. Time correction
factors for a 2 m/s wind on an equal
• Final time in the equal quadrant track is 0.06 seconds
slower than the IAAF standard track simulation.
• Due to smaller radii on the bends. Causes runners to
spend more time on the ground than in the air
• Increases their time.
• An equal quadrant track has longer straights than the
standard track,
to be able to run faster on the straights.
• Does not outweigh the disadvantage that they have on
the bends.
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1. J. Alvarez-Ramirez, An improved Peronnet-Thibault mathematical model of human running performance,
European Journal of Applied Physiology 86 (2002) 617-525.
2. H. Behncke, Small effects in running, Journal of Applied Biomechanics 10 (1994) 270-290.
3. A. Ferro, A. Rivera, and I. Pagola et al., Biomechanical research project at the 77th World
Championships in Athletics Seville 1999, New Studies in Athletics 16 (2001) 25-60.
4. P. R. Greene, Running on flat turns: experiments, theory, and applications, Journal of
Biomechanical Engineering: Transactions of the American Society of Mechanical
Engineers 107 (1985) 96-103.
5. J. B. Keller, A theory of competitive running, Physics Today 26 (1973) 42-47.
6. W. D. McArdle, F. I. Katch, and V. L. Katch, V.L., Exercise Physiology: Energy, Nutrition and Human
Performance, 6th ed., Lippincott, Williams, and Wilkins, Philadelphia, PA, 2007.
7. M. D. Quinn, The effects of wind and altitude in the 200-m sprint, Journal of Applied
Biomechanics 19 (2003) 49-59. 17
8. M. D. Quinn, The effects of wind and altitude in the 400-m sprint, Journal of Sports
Science 22 (2004) 1073-1081.
9. M. R. Spencer and P. B. Gastin, Energy system contribution during 200 to 1500 m
running in highly trained athletes, Medicine and Science in Sports and Exercise 33 (2001)
157-162.
10. R. A. Walpert and C. R. Kyle, Aerodynamics of the human body in sports, Journal of Biomechanics 22 (1989)
1096.
11. A. J. Ward-Smith and P.F. Radford, A mathematical analysis of the 4 x 100m relay,
Journal of Sports Sciences 20 (2002) 369-381.
```