A Problem of Differentiation
Sonia O’Sullivan’s Silver Sortie
Sonia O’Sullivan ran in the 5000m race at the Sydney Olympic Games 2000 and finished second to Romainia’s Gabriella Szabo.
A mathematical equation which describes the last lap can be set up. The mathematical equation which properly describes the last lap would be more complicated than the one I am presenting here. This equation only approximately describes the first half of the last lap.
Distance Equation: s = 0. 013t2 + 5. 6t +4614, 0 <= t < =31, s in metres, t in seconds.
Use this equation to answer the following questions:
Q1. What distance had Sonia O’Sullivan travelled in the race when I started my stop-watch?
Q2. What distance had Sonia O’Sullivan travelled in the race when I stopped my stop-watch?
Q3. What distance did Sonia O’Sullivan travel while my stop-watch was on?
During the last lap the athletes changed the pace at which they had been running over the previous laps. We can now look at the speeds at which Sonia O’Sullivan (and her competitors) travelled at various points during this first half of the last lap – over the approximate 200m.
Use differentiation to obtain the speed (velocity) equation from the distance equation
Speed (velocity) Equation: v = ds/dt = 0. 026t + 5. 6 + 0, 0 < = t < = 31
Q4. What was Sonia O’Sullivan’s speed at the start of the last lap?
Q5. What was Sonia O’Sullivan’s speed half way through the last lap?
Q6. What do you notice about the pace of the race by looking at the two speeds?
We have noticed that the pace of the race has continued to change over the first half of the last lap - according to the mathematical equation which is used to represent the race this is the case and if we look at a recorded video of the race we can also observe the increase in pace visually. We are now going to measure the rate of increase in the speed (velocity). This is called acceleration.
Use differentiation again to obtain the acceleration equation from the speed equation
Acceleration Equation: a = dv/dt (or d2s/dt2 from the distance equation) = 0. 026, 0<= t< = 31
This equation tells us that Sonia O’Sullivan was accelerating
constantly over the first half of the last lap. The rate of acceleration was a change in
speed of 0.
026 metres per second
every second. i.e. an acceleration
of 0. 026ms-2
This mathematical exercise has demonstrated the relationship between the three equations of Distance, Velocity and Acceleration in relation to the first half of the last lap of the 5000m race of Sonia O’Sullivan.
By observing the video of the race it is possible to construct a mathematical expression which describes approximately the second half of the last lap.
It should be noted that the lap times given on the video recording are those of the fastest runner at any stage. These are the times that I used as data for the equations. These equations are approximations for the leader of the race. Prior to the last lap Sonia O’Sullivan was more than 10m off the pace – a distance which required almost 2 seconds to cover. Over the section of the race already described Sonia O’Sullivan’s speeds and acceleration must have been even higher than the results obtained above. It can also be noted that Sonia O’Sullivan frequently ran on an outside lane in order to overtake those ahead. On a full lap each lane out from the inside track adds approximately 6m to the lap length (an extra 1m radius on the curved section leads to 2π extra metres on the track).
An equation which approximately represents the distance covered for the second half of the last lap (200m) is given by
s = 0. 01035t2 + 5. 758t + 4617, 31<= t <= 60
t = 31 gives a distance of 4805. 44435m
t = 60 gives a distance of 4999. 74m (Incredibly, this means that Sonia O’Sullivan did not finish the race! But do not believe it!)
This gives a distance of 194. 29565m travelled during the second half of the lap.
The speed equation is
v = 0. 0207t + 5. 758
t = 31 gives a speed of 6.3997ms-1
t = 60 gives a speed of 7ms-1
The acceleration equation is
a = 0. 0207
It may be noted that a speed of 7ms-1 is
approximately 16mph or 25.2 km/h
Over the final 100m Sonia O’Sullivan used a stride length of 3.88m (approx. 12.8ft.) while the winner, Gabriella Szabo, used a stride length of 3.33m (appox. 10.9ft.).
Try to find a Video of the Race and enjoy viewing it during a Maths class some day!
See this video on Youtube: Women’s 5000m race; Sydney Olympics 2000.
Page created by Neil Hallinan January 2002