### Limited exponential growth: Applications of limited exponential growth models

### The Keller model of sprinting

In biomechanics are two types of mathematical models used for racing: kinematic models based on Newton's second law and models based on energy balance. We discuss the kinematic model of Keller in 1973 that describes a sprint over 100 metres, but can also be used for longer distances with elite athletes.

Keller model of sprinting Like all the kinematic models, the Keller model is based on the relationship \[a(t) = F_{\mathrm{propulsion}}(t)-F_{\mathrm{resistance}}(t)\] where \(a(t)\) is the acceleration of the sprinter at time \(t\), \(F_{\mathrm{propulsion}}\) is the horizontal component of the propulsion force per body weight at time \(t\), and \(F_{\mathrm{resistance}}(t)\) is the force per body weight that the sprinter must overcome at time \(t\) in order to move forward. Keller assumed in his model that the propulsion force is constant and equal to ( \(F\) and that the resistance is mainly determined by the internal resistance in the body that has to be overcome in order to move the limbs (leading to heat generation), and that this resistance increases with the weight of the runner and increases with speed. So this resistance has, contrary to popular belief, nothing to do with air resistance; this would be a separate term you could build into a kinematic model. Keller assumed a linear relationship between the propulsion force and the velocity of the sprinter: \(F_{\mathrm{resistance}}(t)=v(t)/\tau\). This reasoning leads to the following initial value problem \[v'(t)=F-\frac{v(t)}{\tau},\quad v(0)=0\] This equation of limited exponential growth has the following solution \[v(t)=F\cdot \tau\cdot\left(1-e^{\frac{t}{\tau}}\right)\]

Modelling a sprint of Carl Lewis We apply the Keller model to the 100 m sprint of Carl Lewis in the final of the World Athletics Championships 1987 in Rome. Based on his split times per 10 meter you can determine the best matching parameter values via regression. The result is \(F=8.88\;\mathrm{N/kg}\) and \(\tau=1.318\;\mathrm{s}\).

Heck & Ellermeijer (2009) have also used this model to model short sprints of 16-year-olds over a distance of 30 m; they have parameter values \(F=5.988\;\mathrm{N/kg}\) and \(\tau=1.269\;\mathrm{s}\). In particular, the propulsion force is less for a student than for an elite sprinter and probably also the ability to maintain this momentum for some time is less.

Heck, A. & Ellermeijer, T. (2009). Giving students the run or sprinting models. *American Journal of Physics* **77** (11), 1028-1038.